Collection of Infinite Products and Series

   Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

InfProd_1.png

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My interest in infinite products has its origin in the year 2000 in connection with the problem of the electrical field of a
line charge trapped inside a rectangular tube. After I learned that the double product can be solved using
elliptic theta functions I was hooked. The site has been growing ever since, and its focus has been expanded
to include Series as well.

These pages list thousands of expressions like products, sums, relations and limits shown in the following sections:

-  Infinite Products

-  Products involving Theta Functions

-  Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series

-  q-Series

-  special values of EllipticK and EllipticE

-  Series of Hyperbolic Functions

-  Series of CosIntegral

-  Sums involving reciprocal multifactorials or factorials

-  Multiple Sums (lattice sums)

-  diverse Series

-  Series of Logarithms

-  Series of Inverse Tangents ( Arcustangent )

-  Series of Bessel Functions

-  Series of Legendre Polynomials

-  Series of Zeta PolyGamma PolyLog and related

-  Series of Beta Functions

-  Series of Gamma Functions

-  Series involving HarmonicNumber

-  Series involving Hypergeometric Functions

-  some Limits

-  a few Integrals

-  iterated expressions ( Tetration )

-  some properties of ProductLog LerchPhi and PolyLog

{j, n, m} are Integer; {λ, q} > 0 and r are real; {z, InfProd_3.png, InfProd_4.png, InfProd_5.png, InfProd_6.png} may be complex; Γ[a] is  Gamma[a];
InfProd_7.png], InfProd_8.png] are shorthands for the Elliptic Integrals,
sl[x]  cl[x] and ϖ (CurlyPi) denote the lemniscate functions and constant InfProd_9.png;
the notation Σ’ means that the divergent term in multiple sums is excluded.
There are  products that possess pointlike poles, where the denominator of a factor gets zero for certain
values of z. The given domains may not be complete. Some of the expressions are well known,
others may be not; some were found in the depths of the world wide web, the first are derived
from the first product below.

any formula you decide to use should be numerically tested for validity in the users domain  ←

Expressions communicated by other people are marked with (R#) and are referenced below at the bottom.

Infinite Products : ( Back to Top )

InfProd_10.png

This product converges and delivers infinite product representations for many functions if the {a, b, c, d} are
replaced by constants and simple functions of z :

InfProd_11.png

InfProd_12.png

InfProd_13.png

InfProd_14.png

InfProd_15.png

InfProd_16.png

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InfProd_37.gif

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InfProd_49.gif

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Products of two Gammas :

InfProd_63.png

InfProd_64.gif

InfProd_65.png

InfProd_66.png

InfProd_67.png

Partial Fraction Decompositions :

InfProd_68.png

General expression :

InfProd_69.png

some special cases (all having m = 1, except where noted otherwise):

Order 2:

InfProd_70.png

most common  case (a quadratic binomial), with  InfProd_71.png  and  InfProd_72.png:

InfProd_73.png

Decomposition of the general quadratic trinomial applying the shorthands

InfProd_74.png

gives

InfProd_75.png

Order 3:

InfProd_76.png

With 3 abbreviations

InfProd_77.png

the general cubic multinomial  can be written in terms of first order in x:

InfProd_78.png

Order 4:

A simple one :

InfProd_79.png

Decomposition according to the general formula above :

InfProd_80.png

And at last using these 9  subexpressions

InfProd_81.gif

the general multinomial  of 4th order (n = 4) looks  like (expressed again in terms of  first order in x):

InfProd_82.png

  InfProd_83.png as simple function of k :

Decomposition of  finite products into power series

InfProd_84.png

The general case (r determines the start index of the product, the coefficients of x are called r - Stirling Numbers of the first Kind e.g. Oeis: A143493) :

InfProd_85.png

InfProd_86.png

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InfProd_88.png

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InfProd_91.gif

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Special cases with m = 0 :

InfProd_94.png

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InfProd_97.png

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q - Product (0 < q < 1) :

InfProd_99.png

Two kinds of decomposition of the same product :

InfProd_100.gif

The form of the original product returns in the coefficients of the second decomposition sum, so there is a recurring structure, like :

InfProd_101.png

InfProd_102.png

(left hand side taken from the general expression for partial fractions above) equations like given for instance in the next line are valid for different functions f[k] :

InfProd_103.png

If f[k] = InfProd_104.png  then (0 < x)

InfProd_105.png

InfProd_106.png

or if f[k] = InfProd_107.png  then

InfProd_108.png

InfProd_109.png

InfProd_110.png

InfProd_111.png

InfProd_112.png

InfProd_113.png

InfProd_114.png

More Products :

The maximum of the next function is found at InfProd_115.png | f InfProd_116.png)) =  InfProd_117.png,  its zeros on the positive (negative) axis are the odd (even) Integers and in general f[z] = f[-z-1] :

InfProd_118.png

InfProd_119.png

InfProd_120.png

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InfProd_126.gif

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InfProd_139.png

InfProd_140.png

InfProd_141.png

Euler’s product :

InfProd_142.png

InfProd_143.png

The idea for the following product is taken from: Symmetry 2022, 14, 1418. https://doi.org/10.3390/sym14071418 .

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For a large value of m >> n the next product approximates a Gauss function InfProd_153.png with standard deviation InfProd_154.png :

InfProd_155.png

The agreement of the above approximation for m  > InfProd_156.png increases with j and the error is smaller than InfProd_157.png in the intervall (0 < n < 4InfProd_158.png) .

InfProd_159.png

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Products of trig functions :

InfProd_167.png

InfProd_168.gif

Vieta' s product (set x → π/2 in the first product of Cos above)  was maybe the first (anno 1593) documented infinite product.
It was obtained by taking the ratio of the area of the square inscribed  in a circle of radius r to the area of a InfProd_169.png-polygon (built from InfProd_170.png isosceles triangles) inscribed into the same circle
InfProd_171.png,
beginning with n = 2. It ends up at n = ∞, where the area of the polygon is equal to the area of the circle with
InfProd_172.png.

InfProd_173.png

Vieta’s product can be rewritten by pulling the factors 1/2 inside the following square root factors :

InfProd_174.png

and may then be compared to its lemniscatic cousin, where the multiplications inside the square roots are changed to divisions:

InfProd_175.png

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InfProd_220.png

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InfProd_222.png

InfProd_223.gif

Products containing factors built from all primes p:

InfProd_224.png

Special  values :

InfProd_225.png

Special  rational  values :

InfProd_226.png

Products involving Theta Functions    ( Back to Top )

InfProd_227.png is shorthand for EllipticTheta[n, z, q] and InfProd_228.png means EllipticThetaPrime[m, z, q].

InfProd_229.png

Series and Product Representations :

InfProd_230.png

InfProd_231.png

InfProd_232.png

InfProd_233.png

InfProd_234.png

InfProd_235.png

InfProd_236.png

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With  InfProd_238.png[ 0 , q ] a few relations between the theta functions are

InfProd_239.gif

InfProd_240.png

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InfProd_244.png

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These limits give "needles" of height ±1 situated at the extreme values of Cos or Sin respectively (n∼1/(4λ)) :

InfProd_248.png

InfProd_249.png

Approximation of quotients (0.4 < λ) :

InfProd_250.gif

InfProd_251.png

InfProd_252.png

Partial differential equation (pde) :

InfProd_253.png

EllipticThetas with imaginary argument :

InfProd_254.png

With z → 0 this reduces to

InfProd_255.png

Half Lambda :

InfProd_256.png

InfProd_257.png

InfProd_258.png

InfProd_259.png

InfProd_260.png

InfProd_261.png

Double Lambda :

InfProd_262.gif

InfProd_263.png

Other relations :

InfProd_264.png

InfProd_265.png

From an equation involving Eisenstein series InfProd_266.png and InfProd_267.png InfProd_268.png and their connection to theta functions:

InfProd_269.png

Square and square root of q :

InfProd_270.png

InfProd_271.png

Double Argument  (Landen), -  see above for double  InfProd_272.png)  and half  InfProd_273.pngLambda - :

InfProd_274.png

InfProd_275.png

InfProd_276.png

InfProd_277.png

Half Argument :

InfProd_278.png

InfProd_279.png

InfProd_280.png

InfProd_281.png

Proportionality of imaginary and real part of InfProd_282.png ,  InfProd_283.png and  InfProd_284.png with real argument  and imaginary nome:

InfProd_285.png

Derivatives with respect to q :

InfProd_286.png

InfProd_287.png

InfProd_288.png

InfProd_289.png

With help of the pde above is (second or third derivative with respect to the argument z at z = 0) :

InfProd_290.png

For  integrals of the elliptic theta functions scroll to the bottom of :

Table of Definite Integrals

Series of theta functions :

InfProd_292.png

InfProd_293.png

InfProd_294.png

Now InfProd_295.png may be extracted out of the sum because of its periodicity (see table below)
InfProd_296.png
{0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084}

and the remaining sum can be done :

InfProd_297.png

InfProd_298.png

After letting q = InfProd_299.png and some algebra this may be generalized to :

InfProd_300.gif

Infinite sums of elliptic theta functions multiplied with some function f[k] depending on k (as the theta functions are periodic, they may be - up to a sign - be drawn out of the sum) :

InfProd_301.png

Series representation of ratios of theta functions :

InfProd_302.png

InfProd_303.png

InfProd_304.png

The following double products numerically converge best if k ≫ n.

InfProd_305.png

InfProd_306.png

Double product representation of the single theta functions :

InfProd_307.png

InfProd_308.png

InfProd_309.png

InfProd_310.png

If the product over k is done first then products remain containing Tanh or Coth :

InfProd_311.png

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InfProd_315.png

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InfProd_317.png

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The theta functions may be expressed through each other :

InfProd_319.png

and exhibit a kind of double periodicity ({m, n} ∈ Z) :

InfProd_320.png

Products with factors made up of powers or square roots of ratios of integers :

InfProd_321.png

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InfProd_331.gif

InfProd_332.png

InfProd_333.png

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InfProd_336.png

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The next product featuring the lemniscatic constant appears in a similar form compared to the Wallis product directly above (isn’t that amazing ?) :

InfProd_338.png

Both ((*1*) and (*2*)) describe a certain property of members of the family of clover curves given by the polar equation

InfProd_339.png

where r and θ denote the polar coordinates of a point on the curve  (m = 1: Cardioid, m = 2: circle, m = 3: three-leaf-clover, m = 4: lemniscate, m = 5: five-leaf-clover…).
Their principal parts are located inside a cone of width θ = {-π/m, π/m} while the respective arclengths (at r = 1) of the positive half a ‘clover leaf’ can be nicely expressed in form of a Wallis - type product as:

InfProd_340.png

cf.  Hyde: A Wallis product on clovers.

For m up to 5 the arclengths of the half leaves are explicitly:

InfProd_341.png

The quotient of (*2*)/(*1*) is :

InfProd_342.png

and  (*1*)/(*2*) now running from k = 0  gives :

InfProd_343.png

The next Wallis - type product describes the area under a superellipse expresssed by  InfProd_344.png with a = b = 1 inside the first quadrant:

InfProd_345.png

A few values of the product above for integer s from 1 to 6 are

InfProd_346.png

where InfProd_347.png = InfProd_348.png is the real half - period of the Weierstrass elliptic function with invariants g2 = 0, g3 = 1.

InfProd_349.png

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Trigonometric and hyperbolic Products :

InfProd_361.png

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InfProd_375.png

InfProd_376.png

With  m = InverseEllipticNomeQ[Exp[-π λ]] and K[m] = EllipticK[m] :   

InfProd_377.png

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InfProd_379.png

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q - Products :

In the following is ( 0 < q < 1 ) and InfProd_402.png[ 0 , q ] ,   (InfProd_403.png[ 0 , q ] =InfProd_404.png[ 0 , - q ] ) :

InfProd_405.png

InfProd_406.png

m = InverseEllipticNomeQ[q] and K[m] = EllipticK[InverseEllipticNomeQ[q]].

InfProd_407.png

InfProd_408.gif

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m = InverseEllipticNomeQ[q], K[m] = EllipticK[InverseEllipticNomeQ[q]] and E[m] = EllipticE[InverseEllipticNomeQ[q]]:

InfProd_442.png

InfProd_443.png

InfProd_444.png

InfProd_445.png

InverseEllipticNomeQ m[q], K[m[q]] and E[m[q]] expressed through infinite products or theta functions:

InfProd_446.gif

InfProd_447.png

InfProd_448.png

InfProd_449.png

InfProd_450.pngInfProd_451.png and InfProd_452.png can be expressed through m[q] , K[m[q]] and E[m[q]] :

InfProd_453.png

and similarly :

InfProd_454.png

and :

InfProd_455.png

and from combining the above like :

InfProd_456.png

we get :

InfProd_457.png

as q is getting larger than InfProd_458.png the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

InfProd_459.png

or turned the other way round :

InfProd_460.png

InfProd_461.png

If the result of the imaginary transformation doesn't seem right, consider the following points :
• If in the resulting formula a sign change of the imaginary part as function of q occurs under a square root ( at q = Exp[- π / 2] ) then the square root may take the other sign
• Logs with complex arguments may end up on a wrong branch, try replacing Log[...] with Log[...] + n 2 π i

Theta Functions (z = 0, π/4, π/2, 3π/4) expressed through EllipticK and m :

InfProd_462.png InfProd_463.png InfProd_464.png InfProd_465.png
InfProd_466.png InfProd_467.png InfProd_468.png InfProd_469.png
InfProd_470.png InfProd_471.png InfProd_472.png InfProd_473.png
InfProd_474.png InfProd_475.png InfProd_476.png InfProd_477.png
InfProd_478.png InfProd_479.png InfProd_480.png InfProd_481.png
InfProd_482.png InfProd_483.png InfProd_484.png InfProd_485.png
InfProd_486.png InfProd_487.png InfProd_488.png InfProd_489.png
InfProd_490.png InfProd_491.png InfProd_492.png InfProd_493.png

Series expansion of InverseEllipticNomeQ :

InfProd_494.png

With nome q = InfProd_495.png the complementary nome is q’ InfProd_496.png= InfProd_497.png. The following development shows convergence for InfProd_498.png < q, because q’ is rapidly approaching zero with increasing q < 1 :

InfProd_499.png

Connection of InverseEllipticNomeQ to ModularLambda :

InfProd_500.png

InfProd_501.pngInfProd_502.png (n) in Wikipedia (see ' modular lambda function') .

InfProd_503.png

Special values of InfProd_504.png (n) :

InfProd_505.png

Specific Values :

InfProd_506.png

InfProd_507.png

InfProd_508.png

InfProd_509.png

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InfProd_515.gif

InfProd_516.png

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InfProd_518.png

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InfProd_522.png

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A special (remarkable) relation :

InfProd_524.png

InfProd_525.png

InfProd_526.png

InfProd_527.png

InfProd_528.png

EllipticNomeQ :

Series expansion and approximation :

InfProd_529.png

InfProd_530.png

Square and square root of the nomen :

InfProd_531.png

Specific Values :

q[m[#]] = #; from specific values of InverseEllipticNomeQ above like for example q[ m[InfProd_532.png] ]  =  q[ InfProd_533.png]  =InfProd_534.png;

Ramanujans g functions:

InfProd_535.png and InfProd_536.png are Ramanujans g functions, m = InverseEllipticNomeQ[ InfProd_537.png]  (for each n ∈ Integer  the even g and the odd G seem to show a somewhat simpler structure than their counterparts) :

InfProd_538.png

InfProd_539.png

InfProd_540.png

InfProd_541.png

InfProd_542.png

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InfProd_544.png

InfProd_545.png

InfProd_546.png

InfProd_547.png

InfProd_548.png

InfProd_549.png

InfProd_550.png

products with q = InfProd_551.png :

m = InverseEllipticNomeQ[InfProd_552.png],  InfProd_553.png = InfProd_554.png[0, InfProd_555.png] :

InfProd_556.png

InfProd_557.png

InfProd_558.png

InfProd_559.png

special cases :

InfProd_560.png

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InfProd_562.png

InfProd_563.png

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InfProd_565.png

InfProd_566.png

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InfProd_568.png

InfProd_569.png

InfProd_570.png

InfProd_571.png

InfProd_572.png

InfProd_573.png

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InfProd_575.png

InfProd_576.png

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InfProd_579.png

InfProd_580.png

InfProd_581.png

InfProd_582.png

InfProd_583.png

InfProd_584.png

InfProd_585.png

InfProd_586.png

InfProd_587.png

Theta Functions, specific values :

InfProd_588.png

InfProd_589.png

InfProd_590.png

InfProd_591.png

InfProd_592.png

InfProd_593.gif

InfProd_594.gif

InfProd_595.png

InfProd_596.gif

InfProd_597.gif

InfProd_598.png

InfProd_599.png

InfProd_600.png

InfProd_601.png

InfProd_602.png

InfProd_603.png

InfProd_604.png

InfProd_605.png

InfProd_606.png

InfProd_607.png

InfProd_608.png

InfProd_609.png

InfProd_610.png

InfProd_611.png

InfProd_612.png

InfProd_613.png

InfProd_614.png

InfProd_615.png

InfProd_616.png

InfProd_617.png

InfProd_618.png

InfProd_619.png

InfProd_620.gif

InfProd_621.png

Beauty meets well-tempered music…;-) ↓

InfProd_622.png

InfProd_623.png

InfProd_624.png

InfProd_625.png

InfProd_626.png

InfProd_627.png

InfProd_628.png

InfProd_629.png

InfProd_630.png

InfProd_631.png

InfProd_632.png

InfProd_633.png

InfProd_634.png

InfProd_635.png

InfProd_636.png

InfProd_637.png

InfProd_638.png

InfProd_639.png

InfProd_640.png

InfProd_641.png

InfProd_642.png

InfProd_643.png

InfProd_644.png

InfProd_645.png

InfProd_646.png

InfProd_647.png

InfProd_648.png

InfProd_649.png

InfProd_650.png

InfProd_651.png

Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series:   ( Back to Top )

The following double sums numerically converge best if k ≫ n. For numerical checks the finite lower limit should be increased by 1 in case the running index is shifted by -1/2.
ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_652.png means InfProd_653.png.

The double series converge best numerically if k >> n.

InfProd_654.png

InfProd_655.png

InfProd_656.png

InfProd_657.png

InfProd_658.png

InfProd_659.png

InfProd_660.png

InfProd_661.png

InfProd_662.png

InfProd_663.png

InfProd_664.png

InfProd_665.png

InfProd_666.png

InfProd_667.png

InfProd_668.png

InfProd_669.png

InfProd_670.png

More double sums can be found below at 'multiple Sums (Lattice sums)'.

InfProd_671.png

InfProd_672.png

InfProd_673.png

InfProd_674.png

InfProd_675.png

InfProd_676.png

InfProd_677.png

InfProd_678.png

InfProd_679.png

InfProd_680.png

InfProd_681.png

InfProd_682.png

InfProd_683.png

Series involving exponentials :

InfProd_684.png

InfProd_685.png

InfProd_686.png

InfProd_687.png

InfProd_688.png

InfProd_689.png

InfProd_690.png

InfProd_691.png

InfProd_692.png

InfProd_693.png

InfProd_694.png

InfProd_695.png

InfProd_696.png

InfProd_697.png

InfProd_698.png

InfProd_699.png

InfProd_700.png

InfProd_701.png

InfProd_702.png

InfProd_703.gif

InfProd_704.png

InfProd_705.png

InfProd_706.png

InfProd_707.png

Theta functions as series of shifted Gauss functions having equal widths (aren’t these quite remarkable relations? See below in series of hyberbolic functions for a similar phenomenon with shifted Sech and Csch functions connected to lemniscate functions) :

InfProd_708.png

InfProd_709.png

InfProd_710.png

InfProd_711.png

InfProd_712.png

InfProd_713.png

InfProd_714.png

InfProd_715.png

InfProd_716.png

InfProd_717.png

InfProd_718.png

InfProd_719.png

InfProd_720.png

InfProd_721.png

InfProd_722.png

InfProd_723.png

InfProd_724.png

InfProd_725.png

InfProd_726.png

InfProd_727.png

InfProd_728.png

InfProd_729.png

InfProd_730.png

InfProd_731.png

InfProd_732.png

InfProd_733.png

InfProd_734.png

InfProd_735.png

InfProd_736.png

Series involving InfProd_737.png :

InfProd_738.gif

InfProd_739.png

InfProd_740.png

InfProd_741.gif

InfProd_742.png

InfProd_743.png

InfProd_744.png

InfProd_745.png

InfProd_746.png

InfProd_747.png

InfProd_748.png

InfProd_749.png

InfProd_750.png

InfProd_751.png

InfProd_752.png

InfProd_753.png

InfProd_754.png

InfProd_755.png

InfProd_756.png

InfProd_757.png

InfProd_758.png

InfProd_759.png

InfProd_760.png

InfProd_761.png

InfProd_762.png

InfProd_763.png

InfProd_764.png

InfProd_765.png

InfProd_766.png

InfProd_767.png

InfProd_768.png

InfProd_769.png

InfProd_770.png

InfProd_771.png

For low values of m and n this gives (with  PL = InfProd_772.png as shorthand) :

InfProd_773.png

InfProd_774.png InfProd_775.png PL;
InfProd_776.png InfProd_777.png InfProd_778.png
InfProd_779.png InfProd_780.png InfProd_781.png
InfProd_782.png InfProd_783.png InfProd_784.png
InfProd_785.png InfProd_786.png InfProd_787.png

InfProd_788.png

InfProd_789.png

InfProd_790.gif

InfProd_791.png

InfProd_792.png

InfProd_793.png

Series of Stirling numbers :

InfProd_794.png

InfProd_795.png

InfProd_796.png

InfProd_797.png

InfProd_798.png

InfProd_799.png

InfProd_800.png

By the Inversion Theorem for Stirling numbers (S1 ⇔ S2):

InfProd_801.png

InfProd_802.png

Series of trigonometric functions :

InfProd_803.png

InfProd_804.png

InfProd_805.png

InfProd_806.png

InfProd_807.png

InfProd_808.png

InfProd_809.png

InfProd_810.png

InfProd_811.png

InfProd_812.png

InfProd_813.png

InfProd_814.png

InfProd_815.png

InfProd_816.png

InfProd_817.png

InfProd_818.png

InfProd_819.png

InfProd_820.png

InfProd_821.png

InfProd_822.png

InfProd_823.png

InfProd_824.png

InfProd_825.gif

InfProd_826.png

InfProd_827.png

InfProd_828.png

InfProd_829.png

InfProd_830.png

InfProd_831.png

InfProd_832.png

InfProd_833.png

InfProd_834.png

InfProd_835.png

InfProd_836.png

InfProd_837.png

InfProd_838.png

InfProd_839.png

InfProd_840.png

InfProd_841.png

InfProd_842.png

InfProd_843.png

InfProd_844.png

InfProd_845.png

InfProd_846.png

InfProd_847.png

InfProd_848.png

InfProd_849.png

InfProd_850.png

InfProd_851.png

InfProd_852.png

InfProd_853.png

InfProd_854.png

InfProd_855.png

InfProd_856.png

InfProd_857.png

InfProd_858.png

InfProd_859.png

InfProd_860.png

InfProd_861.png

InfProd_862.png

InfProd_863.png

InfProd_864.png

InfProd_865.png

InfProd_866.png

InfProd_867.png

InfProd_868.png

InfProd_869.png

InfProd_870.png

InfProd_871.png

InfProd_872.png

InfProd_873.png

InfProd_874.png

InfProd_875.png

InfProd_876.png

InfProd_877.png

InfProd_878.png

InfProd_879.png

InfProd_880.png

InfProd_881.png

InfProd_882.png

InfProd_883.png

InfProd_884.png

InfProd_885.png

InfProd_886.png

InfProd_887.png

InfProd_888.png

InfProd_889.png

InfProd_890.png

InfProd_891.png

InfProd_892.png

InfProd_893.png

InfProd_894.png

InfProd_895.png

InfProd_896.png

InfProd_897.png

InfProd_898.png

The following approximations hold to about 2% over all a :

InfProd_899.png

Sin[π a] = Cos[π (a - 1/2)] :

InfProd_900.png

InfProd_901.png

InfProd_902.png

InfProd_903.png

InfProd_904.png

InfProd_905.png

InfProd_906.png

InfProd_907.png

InfProd_908.png

InfProd_909.png

InfProd_910.png

InfProd_911.png

InfProd_912.png

InfProd_913.png

InfProd_914.png

InfProd_915.png

InfProd_916.gif

InfProd_917.png

InfProd_918.png

InfProd_919.png

InfProd_920.png

InfProd_921.png

InfProd_922.png

InfProd_923.png

InfProd_924.png

InfProd_925.png

InfProd_926.png

InfProd_927.png

InfProd_928.png

InfProd_929.gif

InfProd_930.png

InfProd_931.png

InfProd_932.png

InfProd_933.png

InfProd_934.png

InfProd_935.png

InfProd_936.png

InfProd_937.png

InfProd_938.png

InfProd_939.png

InfProd_940.png

InfProd_941.gif

InfProd_942.png

InfProd_943.png

InfProd_944.png

InfProd_945.png

InfProd_946.png

InfProd_947.png

InfProd_948.png

InfProd_949.png

InfProd_950.png

InfProd_951.png

InfProd_952.png

InfProd_953.gif

InfProd_954.gif

InfProd_955.png

InfProd_956.png

InfProd_957.png

Amazing identities connecting trigonometric and lemniscate functions:

InfProd_958.png

InfProd_959.png

InfProd_960.png

Special values of trigonometric functions :

Euler :

InfProd_961.png

Sin[π k/n] , n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_962.png

1
InfProd_963.png InfProd_964.png
InfProd_965.png 1 InfProd_966.png
InfProd_967.png InfProd_968.png InfProd_969.png InfProd_970.png
InfProd_971.png InfProd_972.png 1 InfProd_973.png InfProd_974.png
InfProd_975.png InfProd_976.png InfProd_977.png InfProd_978.png InfProd_979.png InfProd_980.png
InfProd_981.png InfProd_982.png InfProd_983.png 1 InfProd_984.png InfProd_985.png InfProd_986.png

Cos[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_987.png

0
InfProd_988.png InfProd_989.png
InfProd_990.png 0 InfProd_991.png
InfProd_992.png InfProd_993.png InfProd_994.png InfProd_995.png
InfProd_996.png InfProd_997.png 0 InfProd_998.png InfProd_999.png
InfProd_1000.png InfProd_1001.png InfProd_1002.png InfProd_1003.png InfProd_1004.png InfProd_1005.png
InfProd_1006.png InfProd_1007.png InfProd_1008.png 0 InfProd_1009.png InfProd_1010.png InfProd_1011.png

Repeated bisection of the angle (2n+1)π inside of trigonometric functions gives (row k = 1 to 6 , column n =  0 to 5):

InfProd_1012.png

1
-1
1
-1
1
-1
InfProd_1013.png
InfProd_1014.png
InfProd_1015.png
InfProd_1016.png
InfProd_1017.png
InfProd_1018.png
InfProd_1019.png
InfProd_1020.png
InfProd_1021.png InfProd_1022.png
InfProd_1023.png
InfProd_1024.png
InfProd_1025.png
InfProd_1026.png
InfProd_1027.png
InfProd_1028.png
InfProd_1029.png
InfProd_1030.png
InfProd_1031.png
InfProd_1032.png
InfProd_1033.png
InfProd_1034.png
InfProd_1035.png
InfProd_1036.png
InfProd_1037.png
InfProd_1038.png
InfProd_1039.png
InfProd_1040.png
InfProd_1041.png
InfProd_1042.png

InfProd_1043.png

0
0
0
0
0
0
InfProd_1044.png
InfProd_1045.png
InfProd_1046.png
InfProd_1047.png
InfProd_1048.png
InfProd_1049.png
InfProd_1050.png
InfProd_1051.png
InfProd_1052.png
InfProd_1053.png
InfProd_1054.png
InfProd_1055.png
InfProd_1056.png
InfProd_1057.png
InfProd_1058.png
InfProd_1059.png
InfProd_1060.png
InfProd_1061.png
InfProd_1062.png
InfProd_1063.png
InfProd_1064.png
InfProd_1065.png
InfProd_1066.png
InfProd_1067.png
InfProd_1068.png
InfProd_1069.png
InfProd_1070.png
InfProd_1071.png
InfProd_1072.png
InfProd_1073.png

Repeated angular bisection of any angle φ  :

InfProd_1074.png

InfProd_1075.png InfProd_1076.png InfProd_1077.png InfProd_1078.png InfProd_1079.png InfProd_1080.png
InfProd_1081.png InfProd_1082.png InfProd_1083.png InfProd_1084.png InfProd_1085.png InfProd_1086.png
InfProd_1087.png InfProd_1088.png InfProd_1089.png InfProd_1090.png InfProd_1091.png InfProd_1092.png
InfProd_1093.png InfProd_1094.png InfProd_1095.png InfProd_1096.png InfProd_1097.png InfProd_1098.png

If Cos[φ] allows a radical expression (see below), then the Cos or Sin of the repeated bisection InfProd_1099.png  also have radical forms as shown exemplarily  in the next two tables (row k = 0 to 3 bisections , angles φ = 0 to 5π/48 in steps of π/48 (columns)):

InfProd_1100.png

sin[0]
0
InfProd_1101.png
InfProd_1102.png
InfProd_1103.png
InfProd_1104.png
InfProd_1105.png
InfProd_1106.png
InfProd_1107.png
InfProd_1108.png
InfProd_1109.png
InfProd_1110.png
sin[0]
0
InfProd_1111.png
InfProd_1112.png
InfProd_1113.png
InfProd_1114.png
InfProd_1115.png
InfProd_1116.png
InfProd_1117.png
InfProd_1118.png
InfProd_1119.png
InfProd_1120.png
sin[0]
0
InfProd_1121.png
InfProd_1122.png
InfProd_1123.png
InfProd_1124.png
InfProd_1125.png
InfProd_1126.png
InfProd_1127.png
InfProd_1128.png
InfProd_1129.png
InfProd_1130.png
sin[0]
0
InfProd_1131.png
InfProd_1132.png
InfProd_1133.png
InfProd_1134.png
InfProd_1135.png
InfProd_1136.png
InfProd_1137.png
InfProd_1138.png
InfProd_1139.png
InfProd_1140.png

InfProd_1141.png

cos[0]
1
InfProd_1142.png
InfProd_1143.png
InfProd_1144.png
InfProd_1145.png
InfProd_1146.png
InfProd_1147.png
InfProd_1148.png
InfProd_1149.png
InfProd_1150.png
InfProd_1151.png
cos[0]
1
InfProd_1152.png
InfProd_1153.png
InfProd_1154.png
InfProd_1155.png
InfProd_1156.png
InfProd_1157.png
InfProd_1158.png
InfProd_1159.png
InfProd_1160.png
InfProd_1161.png
cos[0]
1
InfProd_1162.png
InfProd_1163.png
InfProd_1164.png
InfProd_1165.png
InfProd_1166.png
InfProd_1167.png
InfProd_1168.png
InfProd_1169.png
InfProd_1170.png
InfProd_1171.png
cos[0]
1
InfProd_1172.png
InfProd_1173.png
InfProd_1174.png
InfProd_1175.png
InfProd_1176.png
InfProd_1177.png
InfProd_1178.png
InfProd_1179.png
InfProd_1180.png
InfProd_1181.png

Radicals for Cos[φ] up to π/2 are for example (in steps of π/24) :

InfProd_1182.png

InfProd_1183.png

or in steps of π/10 :

InfProd_1184.png

InfProd_1185.png

Note that 2Cos[ π/5] and 2Cos[2π/5] are equal to Φ (the golden ratio) and to its inverse.

Tan[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_1186.png

ComplexInfinity
InfProd_1187.png InfProd_1188.png
1 ComplexInfinity -1
InfProd_1189.png InfProd_1190.png InfProd_1191.png InfProd_1192.png
InfProd_1193.png InfProd_1194.png ComplexInfinity InfProd_1195.png InfProd_1196.png
InfProd_1197.png InfProd_1198.png InfProd_1199.png InfProd_1200.png InfProd_1201.png InfProd_1202.png
InfProd_1203.png 1 InfProd_1204.png ComplexInfinity InfProd_1205.png -1 InfProd_1206.png

Cot[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_1207.png

0
InfProd_1208.png InfProd_1209.png
1 0 -1
InfProd_1210.png InfProd_1211.png InfProd_1212.png InfProd_1213.png
InfProd_1214.png InfProd_1215.png 0 InfProd_1216.png InfProd_1217.png
InfProd_1218.png InfProd_1219.png InfProd_1220.png InfProd_1221.png InfProd_1222.png InfProd_1223.png
InfProd_1224.png 1 InfProd_1225.png 0 InfProd_1226.png -1 InfProd_1227.png

Some (special) special values :

InfProd_1228.png InfProd_1229.png
InfProd_1230.png InfProd_1231.png
InfProd_1232.png InfProd_1233.png

InfProd_1234.png

InfProd_1235.png InfProd_1236.png
InfProd_1237.png InfProd_1238.png
InfProd_1239.png InfProd_1240.png

InfProd_1241.png

q - Series :   ( Back to Top )

(0 < q < 1); With InfProd_1242.png → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

InfProd_1243.png

InfProd_1244.png

InfProd_1245.png

1
1 1
1 4 1
1 11 11 1
1 26 66 26 1
1 57 302 302 57 1

InfProd_1246.png

InfProd_1247.png

InfProd_1248.png

InfProd_1249.png

The appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_1250.png

There is a small stumble stone in the definition of LerchPhi in the neighbourhood of a = 0: LerchPhi[q, n, a] = InfProd_1251.png, it changes for a = 0 abruptly to a different function InfProd_1252.png.

InfProd_1253.png

InfProd_1254.png

InfProd_1255.png

InfProd_1256.png

InfProd_1257.png

InfProd_1258.png

InfProd_1259.png

InfProd_1260.png

InfProd_1261.png

InfProd_1262.png

InfProd_1263.gif

InfProd_1264.gif

InfProd_1265.png

InfProd_1266.png

InfProd_1267.png

InfProd_1268.png

InfProd_1269.gif

InfProd_1270.png

InfProd_1271.png

InfProd_1272.png

Using an identity from (R4) some series involving the Floor function can be solved  ( InfProd_1273.png[ 0 , q ] is an EllipticTheta function and 0 < q < 1) :

InfProd_1274.png

InfProd_1275.png

InfProd_1276.png

InfProd_1277.png

InfProd_1278.png

InfProd_1279.png

( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):

InfProd_1280.png

InfProd_1281.png

InfProd_1282.png

InfProd_1283.png

InfProd_1284.png

InfProd_1285.png

InfProd_1286.png

InfProd_1287.png

InfProd_1288.png

InfProd_1289.png

InfProd_1290.png

Lambert Type q Series:

InfProd_1291.png

InfProd_1292.png

InfProd_1293.png

InfProd_1294.gif

InfProd_1295.png

InfProd_1296.gif

InfProd_1297.gif

InfProd_1298.png

InfProd_1299.png

InfProd_1300.gif

InfProd_1301.png

InfProd_1302.gif

InfProd_1303.gif

The introduction of QPolyGamma[n, z, q] (nth derivative of the QDigamma function (z, q)) in Mathematica 7 allows expression of

InfProd_1304.png

InfProd_1305.png

InfProd_1306.png

InfProd_1307.png

InfProd_1308.png

InfProd_1309.png

InfProd_1310.png

InfProd_1311.png

InfProd_1312.png

InfProd_1313.png

InfProd_1314.png

InfProd_1315.png

InfProd_1316.png

InfProd_1317.png

InfProd_1318.png

InfProd_1319.png

InfProd_1320.png

InfProd_1321.png

InfProd_1322.png

InfProd_1323.png

InfProd_1324.png

InfProd_1325.png

InfProd_1326.png

InfProd_1327.png

InfProd_1328.png

InfProd_1329.png

InfProd_1330.png

InfProd_1331.png

InfProd_1332.png

InfProd_1333.png

InfProd_1334.png

InfProd_1335.png

InfProd_1336.png

InfProd_1337.png

InfProd_1338.png

InfProd_1339.png

InfProd_1340.png

InfProd_1341.png

The next q - series (q → InfProd_1342.png) are connected to the Eisenstein SeriesInfProd_1343.png like

InfProd_1344.png

InfProd_1345.png

InfProd_1346.png

InfProd_1347.png

InfProd_1348.png

InfProd_1349.png

InfProd_1350.png

InfProd_1351.png

With q = InfProd_1352.png this kind of sum is

InfProd_1353.png

InfProd_1354.png

InfProd_1355.png

InfProd_1356.png

InfProd_1357.png

InfProd_1358.png

InfProd_1359.png

InfProd_1360.png

InfProd_1361.png

InfProd_1362.png

InfProd_1363.png

InfProd_1364.png

InfProd_1365.png

InfProd_1366.png

InfProd_1367.png

InfProd_1368.png

InfProd_1369.png

InfProd_1370.png

InfProd_1371.png

InfProd_1372.png

InfProd_1373.png

InfProd_1374.png

InfProd_1375.png

InfProd_1376.png

InfProd_1377.png

InfProd_1378.png

InfProd_1379.png

InfProd_1380.png

InfProd_1381.png

InfProd_1382.png

InfProd_1383.png

InfProd_1384.png

InfProd_1385.png

InfProd_1386.png

InfProd_1387.png

InfProd_1388.png

InfProd_1389.png

InfProd_1390.png

InfProd_1391.png

InfProd_1392.png

InfProd_1393.png

InfProd_1394.png

InfProd_1395.png

For 0.2 < q is in good approximation :

InfProd_1396.png

InfProd_1397.png

InfProd_1398.png

InfProd_1399.png

InfProd_1400.png

InfProd_1401.png

InfProd_1402.png

InfProd_1403.png

InfProd_1404.png

InfProd_1405.png

InfProd_1406.png

InfProd_1407.png

InfProd_1408.png

InfProd_1409.png

InfProd_1410.png

InfProd_1411.png

InfProd_1412.png

InfProd_1413.png

InfProd_1414.png

InfProd_1415.png

InfProd_1416.png

InfProd_1417.png

InfProd_1418.png

InfProd_1419.png

InfProd_1420.png

InfProd_1421.png

InfProd_1422.png

InfProd_1423.png

InfProd_1424.png

InfProd_1425.png

InfProd_1426.png

InfProd_1427.png

InfProd_1428.png

InfProd_1429.png

InfProd_1430.png

InfProd_1431.png

InfProd_1432.png

InfProd_1433.png

InfProd_1434.png

InfProd_1435.png

InfProd_1436.png

InfProd_1437.png

InfProd_1438.png

InfProd_1439.png

InfProd_1440.png

InfProd_1441.png

InfProd_1442.png

InfProd_1443.png

InfProd_1444.png

InfProd_1445.png

InfProd_1446.png

InfProd_1447.png

InfProd_1448.png

InfProd_1449.png

InfProd_1450.png

InfProd_1451.png

InfProd_1452.png

InfProd_1453.png

InfProd_1454.png

InfProd_1455.png

InfProd_1456.png

InfProd_1457.png

InfProd_1458.png

InfProd_1459.png

InfProd_1460.png

InfProd_1461.png

InfProd_1462.png

InfProd_1463.png

InfProd_1464.png

InfProd_1465.png

InfProd_1466.png

InfProd_1467.png

InfProd_1468.png

InfProd_1469.png

InfProd_1470.png

InfProd_1471.png

InfProd_1472.png

InfProd_1473.png

InfProd_1474.png

InfProd_1475.png

InfProd_1476.png

InfProd_1477.png

InfProd_1478.png

InfProd_1479.png

InfProd_1480.png

InfProd_1481.png

InfProd_1482.png

InfProd_1483.png

InfProd_1484.png

InfProd_1485.png

InfProd_1486.png

InfProd_1487.png

InfProd_1488.png

InfProd_1489.png

InfProd_1490.png

InfProd_1491.png

InfProd_1492.png

InfProd_1493.png

InfProd_1494.png

InfProd_1495.png

InfProd_1496.png

InfProd_1497.png

InfProd_1498.png

InfProd_1499.png

InfProd_1500.png

InfProd_1501.png

InfProd_1502.png

InfProd_1503.png

InfProd_1504.png

InfProd_1505.png

InfProd_1506.png

InfProd_1507.png

InfProd_1508.png

InfProd_1509.png

InfProd_1510.png

InfProd_1511.png

InfProd_1512.png

InfProd_1513.png

InfProd_1514.png

InfProd_1515.png

InfProd_1516.png

InfProd_1517.png

InfProd_1518.png

InfProd_1519.png

InfProd_1520.png

InfProd_1521.gif

InfProd_1522.png

InfProd_1523.png

InfProd_1524.png

InfProd_1525.png

InfProd_1526.gif

InfProd_1527.png

InfProd_1528.png

InfProd_1529.png

InfProd_1530.png

InfProd_1531.png

InfProd_1532.png

InfProd_1533.png

InfProd_1534.png

InfProd_1535.png

InfProd_1536.png

InfProd_1537.png

InfProd_1538.png

InfProd_1539.gif

InfProd_1540.png

InfProd_1541.png

InfProd_1542.png

InfProd_1543.png

InfProd_1544.png

InfProd_1545.png

InfProd_1546.png

InfProd_1547.png

InfProd_1548.png

InfProd_1549.png

InfProd_1550.png

InfProd_1551.png

InfProd_1552.png

InfProd_1553.png

InfProd_1554.png

InfProd_1555.png

InfProd_1556.png

InfProd_1557.png

InfProd_1558.png

InfProd_1559.png

InfProd_1560.png

InfProd_1561.png

other :

InfProd_1562.png

InfProd_1563.png

InfProd_1564.png

InfProd_1565.png

InfProd_1566.png

InfProd_1567.png

InfProd_1568.png

InfProd_1569.png

InfProd_1570.png

InfProd_1571.png

InfProd_1572.png

InfProd_1573.png

InfProd_1574.png

InfProd_1575.png

InfProd_1576.png

InfProd_1577.png

InfProd_1578.png

InfProd_1579.png

InfProd_1580.png

InfProd_1581.png

InfProd_1582.png

QFunction Identities :

InfProd_1583.png

InfProd_1584.png

InfProd_1585.png

InfProd_1586.png

InfProd_1587.png

InfProd_1588.png

InfProd_1589.png

Special values of QPolyGamma :

InfProd_1590.png

InfProd_1591.png

InfProd_1592.png

InfProd_1593.png

InfProd_1594.gif

InfProd_1595.png

InfProd_1596.png

InfProd_1597.png

InfProd_1598.png

InfProd_1599.png

InfProd_1600.png

InfProd_1601.png

InfProd_1602.png

InfProd_1603.png

InfProd_1604.png

InfProd_1605.png

InfProd_1606.png

InfProd_1607.png

InfProd_1608.png

InfProd_1609.png

InfProd_1610.gif

InfProd_1611.png

InfProd_1612.png

InfProd_1613.png

InfProd_1614.png

InfProd_1615.png

InfProd_1616.png

InfProd_1617.png

InfProd_1618.png

InfProd_1619.png

InfProd_1620.png

InfProd_1621.png

InfProd_1622.png

InfProd_1623.png

InfProd_1624.png

InfProd_1625.png

InfProd_1626.png

InfProd_1627.png

InfProd_1628.png

InfProd_1629.png

InfProd_1630.png

InfProd_1631.png

InfProd_1632.png

InfProd_1633.png

InfProd_1634.png

InfProd_1635.png

InfProd_1636.png

InfProd_1637.png

InfProd_1638.png

InfProd_1639.png

InfProd_1640.png

InfProd_1641.png

InfProd_1642.png

InfProd_1643.png

InfProd_1644.png

InfProd_1645.png

InfProd_1646.png

InfProd_1647.png

InfProd_1648.png

InfProd_1649.png

InfProd_1650.png

InfProd_1651.png

InfProd_1652.png

InfProd_1653.png

InfProd_1654.png

InfProd_1655.png

InfProd_1656.png

InfProd_1657.png

InfProd_1658.png

InfProd_1659.png

InfProd_1660.png

InfProd_1661.gif

InfProd_1662.png

InfProd_1663.png

InfProd_1664.png

InfProd_1665.png

With x ∈ Reals is   InfProd_1666.png

InfProd_1667.png
Real Part Imaginary Part
n=1: InfProd_1668.png 0
n=2: InfProd_1669.png InfProd_1670.png
n=3: InfProd_1671.png InfProd_1672.png
n=4: InfProd_1673.png InfProd_1674.png
n=5: InfProd_1675.png InfProd_1676.png
n=6: InfProd_1677.png InfProd_1678.png
n=7: InfProd_1679.png InfProd_1680.png

InfProd_1681.png

special values of EllipticK and EllipticE:   ( Back to Top )

K[m] is EllipticK[m];

InfProd_1682.png

E[m] is EllipticE[m];

InfProd_1683.png

InfProd_1684.png

InfProd_1685.png

InfProd_1686.png

InfProd_1687.png

InfProd_1688.png

InfProd_1689.png

InfProd_1690.png

InfProd_1691.png

InfProd_1692.png

InfProd_1693.png

InfProd_1694.png

InfProd_1695.png

InfProd_1696.png

InfProd_1697.png

Series of Hyperbolic Functions:   ( Back to Top )

InfProd_1698.png

InfProd_1699.png

InfProd_1700.png

InfProd_1701.png

InfProd_1702.gif

InfProd_1703.gif

InfProd_1704.png

InfProd_1705.png

InfProd_1706.png

InfProd_1707.png

ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_1708.png means InfProd_1709.png.

InfProd_1710.png

InfProd_1711.png

InfProd_1712.png

InfProd_1713.png

InfProd_1714.png

InfProd_1715.png

The following series containing λ converge very fast with increasing λ :

InfProd_1716.png

InfProd_1717.png

InfProd_1718.png

InfProd_1719.png

InfProd_1720.png

InfProd_1721.png

InfProd_1722.png

InfProd_1723.png

InfProd_1724.png

InfProd_1725.png

InfProd_1726.png

InfProd_1727.png

InfProd_1728.png

InfProd_1729.png

InfProd_1730.png

InfProd_1731.png

InfProd_1732.png

InfProd_1733.png

InfProd_1734.png

InfProd_1735.png

InfProd_1736.png

InfProd_1737.png

InfProd_1738.png

InfProd_1739.png

InfProd_1740.png

InfProd_1741.png

InfProd_1742.png

InfProd_1743.png

InfProd_1744.png

InfProd_1745.png

InfProd_1746.png

InfProd_1747.png

Some Jacobi elliptic functions :

InfProd_1748.png

InfProd_1749.png

InfProd_1750.png

InfProd_1751.gif

InfProd_1752.png

Special values :

InfProd_1753.png

Hyperbolic series involving the lemniscate functions :

ϖ is the lemniscate constant :

InfProd_1754.png

InfProd_1755.png

Connected to the above series are fast converging expansions of special elliptic functions that equal sl[x] and cl[x], the lemniscate sine and cosine, useful for numeric computation of sl and cl:

InfProd_1756.gif

The first 5 Taylor coefficients of the sum representing sl for increasing index m, see sequence A104203 in OEIS (1, -12, 3024, -4390848, 21224560896,...) :

InfProd_1757.png

0 1.0494342235. -11.900183700. 3023.8648359. -4.3909020798.*^6 2.1224555712.*^10
3 0.99999600967. -12.000008223. 3023.9999831. -4.3908480000.*^6 2.1224560896.*^10
6 1.0000000003. -11.999999999. 3024.0000000. -4.3908480000.*^6 2.1224560896.*^10
9 1.000000000. -12.000000000. 3024.0000000. -4.3908480000.*^6 2.1224560896.*^10

The first 8 Taylor coefficients of the sum representing cl for increasing index m, compare sequence A159600 in OEIS (1, -1, 3, -27, 441, -11529, 442827, -23444883,... but beware,  A159600 excludes a factorInfProd_1758.png needed to obtain the series for cl) :

InfProd_1759.png

0 1.0412730250. -1.0296444931. 3.0214068820. -27.016199247. 441.01703299. -11529.047467. 442827.26751. -2.3444884804.*^7
3 0.99999666957. -0.99999760952. 2.9999982842. -26.999998768. 440.99999912. -11528.999999. 442827.00000. -2.3444883000.*^7
6 1.0000000003. -1.0000000002. 3.0000000001. -27.000000000. 441.00000000. -11529.000000. 442827.00000. -2.3444883000.*^7
9 1.000000000. -1.000000000. 3.0000000000. -27.000000000. 441.00000000. -11529.000000. 442827.00000. -2.3444883000.*^7

Near the real axis the lemniscate functions may be described by Fourier series :

InfProd_1760.gif

InfProd_1761.png

Ramanujan's Cos/Cosh identity :

InfProd_1762.png

The approximation of the the next two series to the lemniscate functions in the area  around the origin improves with the number of included terms in the numerical evaluation (more terms or ‘building blocks‘ cover a larger domain):

InfProd_1763.png

InfProd_1764.png

InfProd_1765.png

InfProd_1766.png

The 'four horsemen of the apocalypse':

InfProd_1767.png

InfProd_1768.png

W A I T !  Have you noticed the beauty of  the series above? The alternating sum of shifted 1/Cosh functions gives the lemniscate cosine, the alternating sum of shifted 1/Sinh functions gives 1/(lemniscate sine)... Isn’t that incredible?

InfProd_1769.png

InfProd_1770.png

InfProd_1771.png

InfProd_1772.png

InfProd_1773.png

InfProd_1774.png

InfProd_1775.png

InfProd_1776.png

InfProd_1777.png

InfProd_1778.png

InfProd_1779.png

InfProd_1780.png

InfProd_1781.png

InfProd_1782.png

The next two series results show a sign flip at every other integer interval in y, shifted by 1/2, due to the complex square root. This switch of sign is taken into account by the factor (-1)^Floor[ y - 1/2 ].

InfProd_1783.png

InfProd_1784.png

InfProd_1785.png

InfProd_1786.png

InfProd_1787.png

InfProd_1788.png

InfProd_1789.png

The QPolyGamma 'monsters' :

InfProd_1790.png

InfProd_1791.png

InfProd_1792.png

InfProd_1793.png

InfProd_1794.png

InfProd_1795.png

InfProd_1796.png

InfProd_1797.png

InfProd_1798.png

InfProd_1799.png

InfProd_1800.png

m = InverseEllipticNomeQ[InfProd_1801.png] :

InfProd_1802.png

InfProd_1803.png

InfProd_1804.png

InfProd_1805.png

InfProd_1806.png

InfProd_1807.png

InfProd_1808.png

InfProd_1809.png

InfProd_1810.png

InfProd_1811.png

InfProd_1812.png

InfProd_1813.png

InfProd_1814.png

InfProd_1815.png

InfProd_1816.png

InfProd_1817.png

InfProd_1818.png

InfProd_1819.png

InfProd_1820.png

InfProd_1821.png

InfProd_1822.png

InfProd_1823.png

InfProd_1824.png

InfProd_1825.png

The next expression uses an idea taken from Weiss, J.D.(2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815 - 2822. http://dx.doi.org/10.4236/am.2014.517269 :

InfProd_1826.png

both series approach - Log[2] from either below (Coth) or above (Tanh) for increasing z .

InfProd_1827.png

InfProd_1828.png

InfProd_1829.png

InfProd_1830.png

InfProd_1831.png

InfProd_1832.png

InfProd_1833.png

InfProd_1834.png

InfProd_1835.png

both series above approach z / (1 - z) from either below (Tanh) or above (Coth) for 1 < x.

InfProd_1836.png

InfProd_1837.png

both series above approach PolyLog[-j,z] from either below (Tanh) or above (Coth) for 1 < x.

InfProd_1838.png

InfProd_1839.png

InfProd_1840.png

InfProd_1841.png

m = InverseEllipticNomeQ[InfProd_1842.png] :

InfProd_1843.png

InfProd_1844.png

The real parts of the next four series are well defined for (-1 < x). The real parts for 0 < x and the imaginary Parts for all x converge very fast for small m.

InfProd_1845.png

InfProd_1846.png

InfProd_1847.png

InfProd_1848.png

InfProd_1849.png

InfProd_1850.png

InfProd_1851.png

InfProd_1852.png

InfProd_1853.png

InfProd_1854.png

InfProd_1855.png

InfProd_1856.png

InfProd_1857.png

InfProd_1858.png

InfProd_1859.png

InfProd_1860.png

InfProd_1861.png

InfProd_1862.png

InfProd_1863.png

InfProd_1864.png

InfProd_1865.png

InfProd_1866.png

InfProd_1867.png

InfProd_1868.png

InfProd_1869.png

InfProd_1870.png

InfProd_1871.png

InfProd_1872.png

InfProd_1873.png

InfProd_1874.png

InfProd_1875.png

InfProd_1876.png

InfProd_1877.png

InfProd_1878.png

InfProd_1879.png

InfProd_1880.png

InfProd_1881.png

m = InverseEllipticNomeQ[InfProd_1882.png] :

InfProd_1883.png

InfProd_1884.png

InfProd_1885.png

InfProd_1886.png

InfProd_1887.png

InfProd_1888.png

Some hyperbolic Identities :

InfProd_1889.png

InfProd_1890.png

InfProd_1891.gif

InfProd_1892.gif

Some Lemniscate Sine and Cosine Identities including derivative and integral:

Periods :

InfProd_1893.png

Dual sibling of the Pythagorean Identity InfProd_1894.png) :

InfProd_1895.png

InfProd_1896.png

Special values :

InfProd_1897.png

InfProd_1898.png

InfProd_1899.png

InfProd_1900.png

InfProd_1901.png

Argument addition formulae :

InfProd_1902.png

Imaginary, negative and double arguments :

InfProd_1903.png

Squares :

InfProd_1904.png

Derivatives and basic integrals :

InfProd_1905.gif

From the argument addition formulae for the lemniscate functions taking  dx small with sl[dx] = dx and cl[dx] = 1 we get the derivative as sl'[x] =InfProd_1906.png (sl[x + dx] - sl[x])/dx = cl[x] (1 + InfProd_1907.png) and likewise for cl'[x].
The integrals may then so be checked by calculating the derivatives of their right hand sides. The expressions show some similarity (duality) with their trigonometric counterparts.

For more integrals of the lemniscate functions scroll to the bottom of :

Table of Indefinite Integrals

The following identities can be verified using the imaginary and symmetry properties of sl and cl given above.

InfProd_1909.png

Complex properties :

InfProd_1910.png

Product approximations to the lemniscate functions (for 0 < m both better than InfProd_1911.png for all x, they  converge more than twice as fast as the already fast converging series given above in the section) :

InfProd_1912.png

The ‘lemniscatic tangent’ is then represented by :

InfProd_1913.png

Lemniscate functions as special cases  InfProd_1914.png of Jacobi elliptic functions  or Neville theta functions:

InfProd_1915.png

InfProd_1916.png

Series of CosIntegral:   ( Back to Top )

InfProd_1917.png

InfProd_1918.png

InfProd_1919.png

InfProd_1920.png

InfProd_1921.png

InfProd_1922.png

InfProd_1923.png

InfProd_1924.png

InfProd_1925.png

InfProd_1926.png

InfProd_1927.png

InfProd_1928.png

InfProd_1929.png

InfProd_1930.png

Sums involving reciprocal multifactorials or factorials:   ( Back to Top )

InfProd_1931.png

InfProd_1932.png

InfProd_1933.png

InfProd_1934.png

InfProd_1935.png

InfProd_1936.png

InfProd_1937.png

For even m = 2j  the right hand side reduces to

InfProd_1938.png

InfProd_1939.png

Higher multifactorials :

In the next equations j designates the number of ! within the multifactorial InfProd_1940.png, γ[a, b] is the lower incomplete Gamma Function Γ[a, 0, b] = InfProd_1941.png .

InfProd_1942.png

InfProd_1943.png

InfProd_1944.png

InfProd_1945.png

InfProd_1946.png

Expressions for still higher orders of k may be obtained by applying InfProd_1947.png to both sides of the equation.

The general expression at x = ±1 is, using recursionally defined coefficients  InfProd_1948.png and InfProd_1949.png (the factor after InfProd_1950.png shows the series result for n = 0) :

InfProd_1951.png

where the recursions

InfProd_1952.png

and

InfProd_1953.png

give the integers InfProd_1954.png and InfProd_1955.png, needed for the calculation of the sum above .

The coefficients obtained with low indices j, n (j counting rows from 1 to 8, n counting columns from 0 to 9) are shown here for the regular sum (+1) and the alternating sum (-1):

InfProd_1956.png

InfProd_1957.png

The numbers InfProd_1958.png and InfProd_1959.png represent the Bell numbers BellB[n, 1] and the complementary Bell numbers BellB[n, -1].

InfProd_1960.png

InfProd_1961.png

InfProd_1962.png

Bell' s polynomes are connected to Stirling numbers of the second kind :

InfProd_1963.png

or  may be calculated as higher derivatives of the exponential function : BellB[n, z] InfProd_1964.png  :

InfProd_1965.png

1 1 1 1
z z z -z
InfProd_1966.png InfProd_1967.png InfProd_1968.png InfProd_1969.png
InfProd_1970.png InfProd_1971.png InfProd_1972.png InfProd_1973.png
InfProd_1974.png InfProd_1975.png InfProd_1976.png InfProd_1977.png
InfProd_1978.png InfProd_1979.png InfProd_1980.png InfProd_1981.png
InfProd_1982.png InfProd_1983.png InfProd_1984.png InfProd_1985.png

InfProd_1986.png

InfProd_1987.png

InfProd_1988.png

A completely crazy series :

InfProd_1989.png

The expression in large brackets represents a polynomial in y and InfProd_1990.png of degree n with integer coefficients.

The sum over s above containing Stirling Numbers of the second kind has this structure:

InfProd_1991.png

and appears in a special combinatorics problem :
It calculates the count of possible ways InfProd_1992.png to distribute a number of a differently colored balls into b indistinguishable boxes, so that each box contains two balls or more (see A008299 in OEIS).
The triangular table below lists, how many configurations exist for a = 4 … 20 balls (row#) to go into b = 2 … a/2 boxes (column#), min. 2 per box,
also known as associated Stirling numbers of the second kind :

Table[BB[a, b], {a, 4, 14}, {b, 2, Floor[a/2]}] // TableForm

3
10
25 15
56 105
119 490 105
246 1918 1260
501 6825 9450 945
1012 22935 56980 17325
2035 74316 302995 190575 10395
4082 235092 1487200 1636635 270270
8177 731731 6914908 12122110 4099095 135135

As an aside:
These numbers can also be computed from sums containing products of binomial coefficients (‘n choose k’), divided possibly by factorials as explained in the following example :

Consider 8 differently colored balls to go into 3 boxes. First find IntegerPartitions  of 8 into 3 integers to see the possible partitions of the balls into the boxes :

IntegerPartitions[8, {3}]

InfProd_1993.png

Select the partitions with every element larger than one → {4, 2, 2} and {3, 3, 2} . Start by putting 4 balls into any emtpy box (8 choose 4, Binomial[8, 4] possibilities)  AND choose 2 balls for another empty box (Binomial[8-4, 2])  AND again 2 balls for the last still empty box (Binomial[8-4-2, 2]). Multiply (AND condition) the binomials. Since two elements are equal (2, 2), divide this term by 2!.
Now add  (OR condition) the term from the next partition: put 3 balls into one emtpy box (Binomial[8, 3])  AND 3 balls in another empty box (Binomial[8-3, 3])  AND again 2 balls into the last empty box (Binomial[8-3-3, 2]) and multiply. Divide also this term by the factorial of the number of equal elements. The first argument of the binomials shows the number of ‘unboxed’ balls still to choose from, the second argument contains the element of the partition. All binomials (each corresponds to a box) of a partition are multiplied and (as the possible arrangements of  boxes with an equal number of balls inside are not distinguished)  divided by the factorial of their multiplicity :

InfProd_1994.png

This result corresponds to triangle entry (a = 8, b = 3) .

The ‘crazy’ series above has been looked at in more detail in : Vigren E .; Dieckmann A .; A New Result in Form of Finite Triple Sums for a Series from Ramanujan' s Notebooks . Symmetry 2022, 14, 1090.

InfProd_1995.png

InfProd_1996.png

Higher powers of the factorial in the denominator :

InfProd_1997.gif

Replace InfProd_1998.png with InfProd_1999.png and InfProd_2000.png  with InfProd_2001.png
as well as  InfProd_2002.png with InfProd_2003.png if all symbolic expressions are to be kept strictly real .
The numerical evaluation  of the results above should work over the whole x - range anyway , cancelling possibly imaginary contributions.

Expressions for still higher orders of k above may be obtained by applying (InfProd_2004.png) to both sides of the equation .

The solution of the general series seems more complicated with a single, but quite messy hypergeometric function, (Table[n,k] with k ≤ 0 gives the empty set {}):

InfProd_2005.png

There are cases, where this bulky result may be reduced to simpler functions :
For 2 < m   and for low j the series is given by Mathematica as a somewhat shorter hypergeometric function than in the general case above :

0 = j :

InfProd_2006.png

0 < j < m:

InfProd_2007.png

m ≤ j :
These can be done at the point x = ±1, where a recursion may be calculated, via the next relation that decomposes a reciprocal multifactorial sum containing InfProd_2008.pnginto sums of powers InfProd_2009.png less than InfProd_2010.png:

InfProd_2011.png

and the recursion for the coefficients c is given by :

InfProd_2012.png

obtained with the Mathematica code in the next line …(adapt the signs for the alternating case):

InfProd_2013.png

For example with n = 6, m = 3 :

InfProd_2014.png

InfProd_2015.png

The InfProd_2016.pngs are then :

InfProd_2017.png

1 0 0 1 1 1 2 6 17
0 1 0 0 1 2 3 5 12
0 0 1 0 0 1 3 6 11

Now with s = {0, 1, 2} there is

InfProd_2018.png

InfProd_2019.png

so that the recursionally defined sum yields :

InfProd_2020.png

InfProd_2021.png

InfProd_2022.png

while Mathematica gives:

InfProd_2023.png

In this way one may derive from the two expressions above reductions of a more complicated hypergeometric function into a series of simpler ones like:

HypergeometricPFQ[{2, 2, 2}, {1, 1, 1, 1, 1}, 1] = 2  HypergeometricPFQ[{}, {1, 1}, 1] + 3 HypergeometricPFQ[{}, {2, 2}, 1] + 3 HypergeometricPFQ[{}, {1, 2}, 1];

The following  identity (valid for all {m, n ∈ N}) expresses the sum without recursion in terms of sums with exponents of k that are reduced from n to ≤ n-m :

InfProd_2024.png

The  highest exponent of InfProd_2025.png appearing in the recursion above is: if (n < m)  then {s = n} else {s = Min[n-m, m-1]}.

For m = 2 the situation is easier, because all sums (0 ≤ n)InfProd_2026.png can be expressed with Bessel functions through the recursion (see above, set again x = 1)
as linear combinations of the one or two lowest results in the regular case with { {j = 0, BesselI[0, 2]}, {j = 1, BesselI[1, 2]} } up to n = 15:

InfProd_2027.png

1 0 1 1 2 5 13 36 109 359 1266 4731 18657 77464 337681 1540381
0 1 0 1 2 4 10 29 90 295 1030 3838 15168 63117 275252 1254801
n InfProd_2028.png
0 BesselI[0,2]
1 BesselI[1,2]
2 BesselI[0,2]
3 BesselI[0,2]+BesselI[1,2]
4 2 BesselI[0,2]+2 BesselI[1,2]
5 5 BesselI[0,2]+4 BesselI[1,2]
6 13 BesselI[0,2]+10 BesselI[1,2]
7 36 BesselI[0,2]+29 BesselI[1,2]
8 109 BesselI[0,2]+90 BesselI[1,2]
9 359 BesselI[0,2]+295 BesselI[1,2]
10 1266 BesselI[0,2]+1030 BesselI[1,2]
11 4731 BesselI[0,2]+3838 BesselI[1,2]
12 18657 BesselI[0,2]+15168 BesselI[1,2]
13 77464 BesselI[0,2]+63117 BesselI[1,2]
14 337681 BesselI[0,2]+275252 BesselI[1,2]
15 1540381 BesselI[0,2]+1254801 BesselI[1,2]

or in the alternating case with { {j = 0, BesselJ[0, 2]}, {j = 1, - BesselJ[1, 2]} }:

InfProd_2029.png

1 0 -1 -1 0 3 9 16 7 -87 -472 -1567 -3375 -216 45927 308107
0 1 0 -1 -2 -2 2 17 54 109 54 -796 -5000 -19499 -52252 -44617
n InfProd_2030.png
0 BesselJ[0,2]
1 -BesselJ[1,2]
2 -BesselJ[0,2]
3 -BesselJ[0,2]+BesselJ[1,2]
4 2 BesselJ[1,2]
5 3 BesselJ[0,2]+2 BesselJ[1,2]
6 9 BesselJ[0,2]-2 BesselJ[1,2]
7 16 BesselJ[0,2]-17 BesselJ[1,2]
8 7 BesselJ[0,2]-54 BesselJ[1,2]
9 -87 BesselJ[0,2]-109 BesselJ[1,2]
10 -472 BesselJ[0,2]-54 BesselJ[1,2]
11 -1567 BesselJ[0,2]+796 BesselJ[1,2]
12 -3375 BesselJ[0,2]+5000 BesselJ[1,2]
13 -216 BesselJ[0,2]+19499 BesselJ[1,2]
14 45927 BesselJ[0,2]+52252 BesselJ[1,2]
15 308107 BesselJ[0,2]+44617 BesselJ[1,2]

A generalization to powers of multifactorials :

InfProd_2031.png

where two recursions are needed:

InfProd_2032.gif

InfProd_2033.png

The series

InfProd_2034.png

converge very fast (the terms k > 3 contribute less than InfProd_2035.png) . For m = 3  the terms give :

InfProd_2036.png

With n = 0 the first sum and (InfProd_2037.png- the second sum) are very close to the  Pomerance Number  1.25002143347...(A100085 in Oeis).

Multiple Sums (lattice sums) :   ( Back to Top )

Factorial :

InfProd_2038.png

InfProd_2039.png

InfProd_2040.png

InfProd_2041.png

InfProd_2042.png

InfProd_2043.png

InfProd_2044.png

InfProd_2045.png

InfProd_2046.png

InfProd_2047.png

InfProd_2048.png

InfProd_2049.png

InfProd_2050.png

InfProd_2051.png

InfProd_2052.png

InfProd_2053.png

InfProd_2054.png

InfProd_2055.png

InfProd_2056.png

InfProd_2057.png

InfProd_2058.png

Borwein' s formula :

InfProd_2059.png

Benson' s formula:

InfProd_2060.png

already the sum for all {m, n} < 5 gives a very good approximation .

InfProd_2061.png

InfProd_2062.png

InfProd_2063.png

InfProd_2064.png

InfProd_2065.png

InfProd_2066.png

InfProd_2067.png

InfProd_2068.png

InfProd_2069.png

InfProd_2070.png

InfProd_2071.png

InfProd_2072.png

InfProd_2073.png

InfProd_2074.png

InfProd_2075.png

InfProd_2076.png

InfProd_2077.png

InfProd_2078.png

InfProd_2079.png

The approximations for the first two sums in the two following lines are for 2 < m better than 10^-10, the Cot Csc term corresponds to the dominant j = 0 term in the double sum.

InfProd_2080.png

InfProd_2081.png

Double series involving lemniscate sine and cosine :

InfProd_2082.png

InfProd_2083.png

InfProd_2084.png

Both sums above show only asymmetric convergence, i.e.  the range of summation index i must be much larger than the one of j, the convergence of the imaginary part is faster.

InfProd_2085.png

Some values of Eisenstein series :

InfProd_2086.png

InfProd_2087.gif

The coefficients of InfProd_2088.png written as hn[4s], (s = 2, 3,…) are called Hurwitz numbers and may be calculated by the following recurrence:

InfProd_2089.png

InfProd_2090.png

or directly by :

InfProd_2091.png

InfProd_2092.png

InfProd_2093.png

For s = 6, 10, 14, … the above right hand side of the equation above yields zero.

InfProd_2094.png

InfProd_2095.png

For s = 6, 10, 14, …  at x = 1 the right hand side of the equation above yields zero.

InfProd_2096.png

InfProd_2097.png

InfProd_2098.png

InfProd_2099.png

InfProd_2100.png

InfProd_2101.png

InfProd_2102.png

InfProd_2103.png

InfProd_2104.png

InfProd_2105.png

InfProd_2106.png

InfProd_2107.png

InfProd_2108.png

InfProd_2109.png

InfProd_2110.png

some of the next series appear also in electrostatic problems, see
Vigren E.; Dieckmann A.; Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. Symmetry 2020, 12, 1040.

InfProd_2111.png

InfProd_2112.png

InfProd_2113.png

InfProd_2114.png

InfProd_2115.png

InfProd_2116.png

InfProd_2117.png

The encounter of lemniscate functions with electrostatics suggests a tribute to C.F.Gauss, who paved the way for a better understanding of so many problems.

InfProd_2118.png

InfProd_2119.png

InfProd_2120.png

The next double Series converges (asymmetric convergence) best numerically if i >> j.

InfProd_2121.png

Double Series involving the Zeta function :

InfProd_2122.png

InfProd_2123.png

InfProd_2124.png

InfProd_2125.gif

InfProd_2126.png

InfProd_2127.png

InfProd_2128.png

Multiple Series involving the Zeta function :

InfProd_2129.gif

InfProd_2130.png

For a few values of s = 2 to 6 (rows) and m = 1 to s - 1 (columns) the expansions of the right hand side above are given :

InfProd_2131.png
Zeta[3] InfProd_2132.png
InfProd_2133.png InfProd_2134.png InfProd_2135.png
Zeta[5] InfProd_2136.png InfProd_2137.png InfProd_2138.png
InfProd_2139.png InfProd_2140.png InfProd_2141.png InfProd_2142.png InfProd_2143.png

InfProd_2144.png

For a few values of s = 2 to 7 (rows) and m = 1 to s - 1 (columns) the expansions of the right hand side above are given (for m = 1 replace the returned 0’s with Zeta[s]) :

InfProd_2145.png
Zeta[3] InfProd_2146.png
InfProd_2147.png Zeta[3] InfProd_2148.png
Zeta[5] InfProd_2149.png InfProd_2150.png InfProd_2151.png
InfProd_2152.png Zeta[5] InfProd_2153.png InfProd_2154.png InfProd_2155.png
Zeta[7] InfProd_2156.png InfProd_2157.png InfProd_2158.png InfProd_2159.png InfProd_2160.png

The notation InfProd_2161.pngmeans that the divergent term 1/0 is excluded :

InfProd_2162.png

For a few values of  s = 1 to 6 (rows) and m = 1 to 5 (columns)  the expansions of the right hand side above are given :

Log[2] InfProd_2163.png InfProd_2164.png InfProd_2165.png InfProd_2166.png
InfProd_2167.png InfProd_2168.png InfProd_2169.png InfProd_2170.png InfProd_2171.png
InfProd_2172.png InfProd_2173.png InfProd_2174.png InfProd_2175.png InfProd_2176.png
InfProd_2177.png InfProd_2178.png InfProd_2179.png InfProd_2180.png InfProd_2181.png
InfProd_2182.png InfProd_2183.png InfProd_2184.png InfProd_2185.png InfProd_2186.png
InfProd_2187.png InfProd_2188.png InfProd_2189.png InfProd_2190.png InfProd_2191.png

InfProd_2192.png

where the number of numerically equal summands  InfProd_2193.png) in the first sum is counted by the multiplicity factor in large brackets within the second sum;

InfProd_2194.png

InfProd_2195.png

s controls the number of ‘slots’ inside the curly brackets of the Hypergeometric function, e.g. :

s = 4; HypergeometricPFQ[Join[Table[1, {k, 1, s}], {m}], Table[2, {k, 1, s}], -1]

InfProd_2196.png

For a few values of  s = 1 to 6 (rows) and m = 1 to 5 (columns) the expansions of the hypergeometric result above are given :

Table[HypergeometricPFQ[Join[Table[1, {k, 1, s}], {m}], Table[2, {k, 1, s}], -1], {s, 1, 6}, {m, 1, 5}] // FullSimplify // PowerExpand // Expand // TableForm

Log[2] InfProd_2197.png InfProd_2198.png InfProd_2199.png InfProd_2200.png
InfProd_2201.png Log[2] InfProd_2202.png InfProd_2203.png InfProd_2204.png
InfProd_2205.png InfProd_2206.png InfProd_2207.png InfProd_2208.png InfProd_2209.png
InfProd_2210.png InfProd_2211.png InfProd_2212.png InfProd_2213.png InfProd_2214.png
InfProd_2215.png InfProd_2216.png InfProd_2217.png InfProd_2218.png InfProd_2219.png
InfProd_2220.png InfProd_2221.png InfProd_2222.png InfProd_2223.png InfProd_2224.png

InfProd_2225.png

InfProd_2226.png

For a few values of s = 1 to 7 (rows) and m = 1 to 5 (columns)  the  results for the series above are given using a summation, that avoids ‘indeterminate’ answers.
Now the multiplicity of numerically equal summands  InfProd_2227.png) is determined by Binomial[k - 1, m - 1]:

InfProd_2228.png

-Log[2] InfProd_2229.png InfProd_2230.png InfProd_2231.png InfProd_2232.png
InfProd_2233.png InfProd_2234.png InfProd_2235.png InfProd_2236.png InfProd_2237.png
InfProd_2238.png InfProd_2239.png InfProd_2240.png InfProd_2241.png InfProd_2242.png
InfProd_2243.png InfProd_2244.png InfProd_2245.png InfProd_2246.png InfProd_2247.png
InfProd_2248.png InfProd_2249.png InfProd_2250.png InfProd_2251.png InfProd_2252.png
InfProd_2253.png InfProd_2254.png InfProd_2255.png InfProd_2256.png InfProd_2257.png
InfProd_2258.png InfProd_2259.png InfProd_2260.png InfProd_2261.png InfProd_2262.png

InfProd_2263.png

For the lowest values of s and m this sum is:

InfProd_2264.png

and

InfProd_2265.png

For the lowest values of s and m this sum is:

InfProd_2266.png

InfProd_2267.png

For the lowest values of s and m the sum is :

InfProd_2268.png

InfProd_2269.png

For the lowest values of s and m this sum is :

InfProd_2270.png

Many of the series found in this table are connected to a 'lattice version' like (try it!) :

InfProd_2271.png

For instance  (find an identity with start index 0, replace  kInfProd_2272.png ,  insert the Gammas, the j - 1 factorial  and sum over all i's) :

InfProd_2273.png

For a series  InfProd_2274.png starting with index 1 it is a little more involved, because the first lattice summand will be InfProd_2275.png :

InfProd_2276.png

So (for example) the lattice version of

InfProd_2277.png

InfProd_2278.png

Lattice q - sums :

InfProd_2279.png

InfProd_2280.png

InfProd_2281.png

Following ideas of (R4) look at the m-dimensional lattice sum

InfProd_2282.png

where the summands characterized by  InfProd_2283.png = k occur with a certain multiplicity InfProd_2284.png given by

InfProd_2285.png

This tells the number of ways to express k as a sum of m integers InfProd_2286.png (how often a certain term ' k' occurs) ,
for instance  (k = 3, m = 2; 4 ways) : 3 = 0 + 3 or 3 + 0 or 1 + 2 or 2 + 1.
Then the lattice sum can be reduced to a single sum like

InfProd_2287.png

which gives after multiplication with (m - 1)! and evaluation

InfProd_2288.png

This identity can be used to successively get values of InfProd_2289.png. It  determines InfProd_2290.png at m = 3:

InfProd_2291.png

Expanding the sum over (k+2) shows InfProd_2292.png and InfProd_2293.png, and if it is assumed that InfProd_2294.png = 1/2 (regularization), then it follows that InfProd_2295.png = - 1/4.

For m = n + 2 the sum reads :

InfProd_2296.png

The product inside the sum may be decomposed into a double series of StirlingS1 numbers :

InfProd_2297.png

2+k 2+k
InfProd_2298.png InfProd_2299.png
InfProd_2300.png InfProd_2301.png
InfProd_2302.png InfProd_2303.png
InfProd_2304.png InfProd_2305.png

Isolating the term in (**) with the highest exponent (set the stop index in the sum over j in the table to n - 1) now allows a recursive calculation of the InfProd_2306.pnglike:

InfProd_2307.png

InfProd_2308.png

Shown above are the InfProd_2309.png for n from 0 to 10 together with the results of corresponding Mathematica sums employing ' Abel' regularization as well as the symbolic HurwitzZeta given at the start of the paragraph.

InfProd_2310.png

diverse Series :   ( Back to Top )

InfProd_2311.png

InfProd_2312.png

InfProd_2313.png

InfProd_2314.png

InfProd_2315.png

InfProd_2316.png

InfProd_2317.png

InfProd_2318.png

InfProd_2319.png

InfProd_2320.png

InfProd_2321.png

The appearing of n or m as summation stop index implies n, m ∈ N.

InfProd_2322.png

InfProd_2323.png

InfProd_2324.png

InfProd_2325.png

InfProd_2326.png

InfProd_2327.png

InfProd_2328.png

InfProd_2329.png

InfProd_2330.png

InfProd_2331.png

InfProd_2332.png

InfProd_2333.png

InfProd_2334.png

( Zeta[n - 1] - Zeta[n] = Zeta[n - 1, 0] - Zeta[n, 0] = Zeta[n - 1, 1] - Zeta[n, 1] = Zeta[n - 1, 2] - Zeta[n, 2],   Zeta[n,1] = Zeta[n,0] = Zeta[n] )

InfProd_2335.png

InfProd_2336.png

InfProd_2337.png

InfProd_2338.png

InfProd_2339.png

InfProd_2340.png

InfProd_2341.png

InfProd_2342.png

InfProd_2343.png

InfProd_2344.png

InfProd_2345.png

InfProd_2346.png

InfProd_2347.png

InfProd_2348.png

InfProd_2349.png

InfProd_2350.png

InfProd_2351.png

InfProd_2352.png

InfProd_2353.png

InfProd_2354.png

InfProd_2355.gif

InfProd_2356.png

InfProd_2357.png

InfProd_2358.png

InfProd_2359.png

InfProd_2360.png

InfProd_2361.png

InfProd_2362.png

Sum of the inverse m - gonal numbers :

InfProd_2363.png

InfProd_2364.png

Values of the series for the first m :

InfProd_2365.png

The sum of  integer powers  of the inverse m - gonal numbers :

InfProd_2366.png

The quadratic case m = 4, where the formula above gives an indeterminate result,  may be calculated as :

InfProd_2367.png

k - nomial triangles:

Generate the  k - nomial triangle  as coefficents of InfProd_2368.png:
(the mth coefficient in the nth row  gives the frequency of the sum of points with value m + n - 2, shown after a throw of n - 1  fair k-sided dice; displayed are the cases k = {2 bi-, 3 tri-, 4 quadrinomial}, up to n = 5)
InfProd_2369.gif

other series:

The sum  InfProd_2370.png  gives following results for some rational s = p/q :

InfProd_2371.png

InfProd_2372.png

InfProd_2373.png

InfProd_2374.png

InfProd_2375.gif

InfProd_2376.png

InfProd_2377.png

InfProd_2378.png

InfProd_2379.png

InfProd_2380.png

InfProd_2381.png

InfProd_2382.png

InfProd_2383.png

InfProd_2384.png

InfProd_2385.png

InfProd_2386.png

InfProd_2387.png

InfProd_2388.png

InfProd_2389.png

InfProd_2390.png

InfProd_2391.png

InfProd_2392.png

InfProd_2393.png

InfProd_2394.png

InfProd_2395.png

InfProd_2396.png

InfProd_2397.png

InfProd_2398.png

InfProd_2399.png

InfProd_2400.png

InfProd_2401.png

InfProd_2402.png

InfProd_2403.png

InfProd_2404.png

InfProd_2405.png

InfProd_2406.png

InfProd_2407.png

InfProd_2408.png

This sum alternates between ± π  for z ∈ N :

InfProd_2409.png

In the following 4 expressions b =InfProd_2410.png :

InfProd_2411.png

InfProd_2412.png

InfProd_2413.png

InfProd_2414.png

InfProd_2415.png

InfProd_2416.png

InfProd_2417.png

InfProd_2418.png

InfProd_2419.png

InfProd_2420.png

InfProd_2421.png

The next three expressions contain s = InfProd_2422.png and t = InfProd_2423.png:

InfProd_2424.png

InfProd_2425.png

InfProd_2426.png

InfProd_2427.png

InfProd_2428.png

InfProd_2429.png

InfProd_2430.png

InfProd_2431.png

InfProd_2432.png

InfProd_2433.png

InfProd_2434.png

InfProd_2435.png

InfProd_2436.png

InfProd_2437.png

InfProd_2438.png

InfProd_2439.png

InfProd_2440.png

InfProd_2441.png

InfProd_2442.png

InfProd_2443.png

InfProd_2444.png

InfProd_2445.png

During a visit in London Leibniz was asked by Huygens to evaluate the sum InfProd_2446.png. He knew about partial fraction decomposition (see next line) and solved it. (Hirsch, der berühmte Herr Leibniz : eine Biographie)

InfProd_2447.gif

InfProd_2448.png

InfProd_2449.png

InfProd_2450.png

InfProd_2451.png

The sum of  integer powers  of the inverse trigonal numbers :

InfProd_2452.png

InfProd_2453.png

InfProd_2454.png

InfProd_2455.png

InfProd_2456.png

InfProd_2457.png

InfProd_2458.png

InfProd_2459.png

InfProd_2460.png

InfProd_2461.png

InfProd_2462.png

InfProd_2463.png

InfProd_2464.png

InfProd_2465.png

InfProd_2466.png

InfProd_2467.png

InfProd_2468.png

InfProd_2469.png

InfProd_2470.png

InfProd_2471.png

InfProd_2472.png

InfProd_2473.png

InfProd_2474.png

InfProd_2475.png

InfProd_2476.png

InfProd_2477.png

InfProd_2478.png

InfProd_2479.png

InfProd_2480.png

InfProd_2481.png

InfProd_2482.png

InfProd_2483.png

InfProd_2484.png

InfProd_2485.png

InfProd_2486.png

InfProd_2487.png

InfProd_2488.png

InfProd_2489.png

InfProd_2490.png

Series of Logarithms :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_2491.png

InfProd_2492.png

InfProd_2493.png

InfProd_2494.png

InfProd_2495.png

InfProd_2496.png

InfProd_2497.png

InfProd_2498.png

InfProd_2499.png

InfProd_2500.png

InfProd_2501.png

InfProd_2502.png

InfProd_2503.png

InfProd_2504.png

InfProd_2505.png

InfProd_2506.png

InfProd_2507.png

InfProd_2508.png

InfProd_2509.png

InfProd_2510.png

InfProd_2511.png

InfProd_2512.png

InfProd_2513.png

InfProd_2514.png

InfProd_2515.png

InfProd_2516.png

InfProd_2517.png

InfProd_2518.png

InfProd_2519.png

InfProd_2520.png

InfProd_2521.png

InfProd_2522.png

InfProd_2523.png

InfProd_2524.png

InfProd_2525.png

InfProd_2526.png

InfProd_2527.gif

InfProd_2528.png

InfProd_2529.png

InfProd_2530.png

InfProd_2531.png

InfProd_2532.png

InfProd_2533.png

InfProd_2534.png

InfProd_2535.png

InfProd_2536.gif

InfProd_2537.png

InfProd_2538.png

InfProd_2539.png

InfProd_2540.png

InfProd_2541.png

InfProd_2542.png

InfProd_2543.png

InfProd_2544.png

InfProd_2545.png

InfProd_2546.png

InfProd_2547.png

InfProd_2548.png

InfProd_2549.png

InfProd_2550.png

InfProd_2551.png

InfProd_2552.png

InfProd_2553.png

InfProd_2554.png

InfProd_2555.png

InfProd_2556.png

InfProd_2557.png

InfProd_2558.png

InfProd_2559.png

InfProd_2560.png

InfProd_2561.png

InfProd_2562.png

InfProd_2563.png

InfProd_2564.png

InfProd_2565.png

InfProd_2566.png

InfProd_2567.png

InfProd_2568.png

InfProd_2569.png

InfProd_2570.png

InfProd_2571.png

InfProd_2572.png

InfProd_2573.png

InfProd_2574.png

InfProd_2575.png

InfProd_2576.png

InfProd_2577.png

InfProd_2578.png

InfProd_2579.png

InfProd_2580.png

InfProd_2581.png

InfProd_2582.png

InfProd_2583.png

InfProd_2584.png

InfProd_2585.png

InfProd_2586.png

InfProd_2587.png

InfProd_2588.png

InfProd_2589.png

InfProd_2590.png

InfProd_2591.png

InfProd_2592.png

InfProd_2593.png

InfProd_2594.png

InfProd_2595.png

InfProd_2596.png

InfProd_2597.png

InfProd_2598.png

InfProd_2599.png

InfProd_2600.png

InfProd_2601.png

Next is the 'Fountain' function, plot it in the range of -50 < z < 10 with parameter values of a between -3 and 1 !

InfProd_2602.png

InfProd_2603.png

InfProd_2604.png

InfProd_2605.gif

InfProd_2606.png

InfProd_2607.png

InfProd_2608.png

InfProd_2609.png

InfProd_2610.png

InfProd_2611.png

InfProd_2612.png

InfProd_2613.png

InfProd_2614.png

InfProd_2615.png

InfProd_2616.png

InfProd_2617.png

InfProd_2618.png

InfProd_2619.png

InfProd_2620.png

InfProd_2621.png

InfProd_2622.png

InfProd_2623.png

InfProd_2624.png

InfProd_2625.png

InfProd_2626.png

InfProd_2627.png

InfProd_2628.png

InfProd_2629.png

InfProd_2630.png

InfProd_2631.png

InfProd_2632.png

InfProd_2633.png

InfProd_2634.png

InfProd_2635.png

InfProd_2636.png

InfProd_2637.png

InfProd_2638.png

InfProd_2639.png

some of the next series appear also in electrostatic problems, see
Vigren E.; Dieckmann A. ; Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. Symmetry 2020, 12, 1040.

InfProd_2640.png

Lattice Sums :

InfProd_2641.png

InfProd_2642.png

InfProd_2643.png

Exchange x ⇔ y in previous expression :

InfProd_2644.png

derived from above series :

InfProd_2645.png

InfProd_2646.png

The double sum below leads to Green' s function for the Laplace Operator in two dimensions inside a rectangle with sides a and b, the point source being located at xq, yq :

InfProd_2647.png

InfProd_2648.png

Series over prime numbers :

InfProd_2649.png

InfProd_2650.png

InfProd_2651.png

Series of Inverse Tangents ( Arcustangent ) :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_2652.png

InfProd_2653.png

InfProd_2654.png

InfProd_2655.png

InfProd_2656.png

InfProd_2657.png

InfProd_2658.png

InfProd_2659.png

InfProd_2660.png

InfProd_2661.png

InfProd_2662.png

InfProd_2663.png

InfProd_2664.png

InfProd_2665.png

InfProd_2666.png

InfProd_2667.png

InfProd_2668.png

InfProd_2669.png

InfProd_2670.png

InfProd_2671.png

InfProd_2672.gif

InfProd_2673.png

InfProd_2674.png

InfProd_2675.png

LogGamma[z] is used, because it has a simpler branch strucure than Log[Gamma[z]] and avoids many discontinuities.

InfProd_2676.png

InfProd_2677.png

InfProd_2678.png

InfProd_2679.png

InfProd_2680.png

InfProd_2681.png

InfProd_2682.png

InfProd_2683.png

InfProd_2684.png

InfProd_2685.png

InfProd_2686.png

InfProd_2687.png

InfProd_2688.png

InfProd_2689.png

InfProd_2690.png

InfProd_2691.png

InfProd_2692.png

InfProd_2693.png

InfProd_2694.png

InfProd_2695.png

InfProd_2696.png

InfProd_2697.png

InfProd_2698.png

InfProd_2699.png

InfProd_2700.png

InfProd_2701.png

InfProd_2702.png

InfProd_2703.png

InfProd_2704.png

InfProd_2705.png

InfProd_2706.png

InfProd_2707.png

InfProd_2708.png

InfProd_2709.png

InfProd_2710.png

InfProd_2711.png

InfProd_2712.png

InfProd_2713.png

InfProd_2714.png

InfProd_2715.png

InfProd_2716.png

InfProd_2717.png

InfProd_2718.png

InfProd_2719.png

InfProd_2720.png

InfProd_2721.png

InfProd_2722.png

InfProd_2723.png

Lattice Sums :

InfProd_2724.png

InfProd_2725.png

Some ArcTan Identities :

InfProd_2726.gif

InfProd_2727.png

InfProd_2728.png

InfProd_2729.png

InfProd_2730.png

Special values :

InfProd_2731.png

Series of Bessel Functions :   ( Back to Top )

InfProd_2732.png

InfProd_2733.png

InfProd_2734.png

InfProd_2735.png

InfProd_2736.png

InfProd_2737.png

InfProd_2738.png

InfProd_2739.png

InfProd_2740.png

InfProd_2741.png

InfProd_2742.png

InfProd_2743.png

InfProd_2744.png

InfProd_2745.png

InfProd_2746.png

InfProd_2747.png

InfProd_2748.png

InfProd_2749.png

InfProd_2750.png

InfProd_2751.png

InfProd_2752.png

InfProd_2753.png

InfProd_2754.png

InfProd_2755.png

InfProd_2756.png

InfProd_2757.png

InfProd_2758.png

InfProd_2759.png

InfProd_2760.png

InfProd_2761.png

InfProd_2762.png

InfProd_2763.png

InfProd_2764.png

InfProd_2765.png

InfProd_2766.png

InfProd_2767.png

InfProd_2768.png

InfProd_2769.png

InfProd_2770.png

InfProd_2771.png

InfProd_2772.png

InfProd_2773.png

InfProd_2774.png

InfProd_2775.png

InfProd_2776.png

InfProd_2777.png

InfProd_2778.png

InfProd_2779.png

InfProd_2780.png

InfProd_2781.png

InfProd_2782.png

InfProd_2783.png

InfProd_2784.png

For numerical tests replace every ∞ in the results with the same (large enough) number.

InfProd_2785.png

InfProd_2786.png

InfProd_2787.png

InfProd_2788.png

InfProd_2789.png

InfProd_2790.png

InfProd_2791.png

InfProd_2792.png

InfProd_2793.png

InfProd_2794.png

InfProd_2795.png

InfProd_2796.png

InfProd_2797.png

InfProd_2798.png

InfProd_2799.png

InfProd_2800.png

InfProd_2801.png

InfProd_2802.png

InfProd_2803.png

InfProd_2804.png

InfProd_2805.png

InfProd_2806.png

InfProd_2807.png

InfProd_2808.png

InfProd_2809.png

InfProd_2810.png

InfProd_2811.png

InfProd_2812.png

InfProd_2813.png

InfProd_2814.png

InfProd_2815.png

InfProd_2816.png

InfProd_2817.png

InfProd_2818.png

InfProd_2819.png

InfProd_2820.png

InfProd_2821.png

InfProd_2822.png

InfProd_2823.png

InfProd_2824.png

InfProd_2825.png

InfProd_2826.png

InfProd_2827.png

InfProd_2828.png

InfProd_2829.png

InfProd_2830.png

InfProd_2831.png

Cases of Neumann' s addition theorem :

InfProd_2832.png

InfProd_2833.png

Cases of Graf' s addition theorem :

InfProd_2834.png

InfProd_2835.png

InfProd_2836.png

InfProd_2837.png

InfProd_2838.png

InfProd_2839.png

InfProd_2840.png

InfProd_2841.png

InfProd_2842.png

InfProd_2843.png

InfProd_2844.png

InfProd_2845.png

InfProd_2846.png

InfProd_2847.gif

InfProd_2848.png

InfProd_2849.png

InfProd_2850.png

InfProd_2851.png

InfProd_2852.png

InfProd_2853.png

Series of Legendre Polynomials :   ( Back to Top )

InfProd_2854.png

InfProd_2855.png

InfProd_2856.png

InfProd_2857.png

InfProd_2858.png

InfProd_2859.png

InfProd_2860.png

InfProd_2861.png

InfProd_2862.png

InfProd_2863.png

InfProd_2864.png

InfProd_2865.png

InfProd_2866.png

InfProd_2867.png

InfProd_2868.png

InfProd_2869.png

InfProd_2870.png

InfProd_2871.png

InfProd_2872.png

InfProd_2873.png

InfProd_2874.png

InfProd_2875.png

InfProd_2876.png

InfProd_2877.png

InfProd_2878.png

Laguerre Polynomials:

InfProd_2879.png

InfProd_2880.png

Laguerre Polynomials with negative Index

InfProd_2881.png

Series of Jacobi Polynomials :

InfProd_2882.png

InfProd_2883.png

Series of Hermite Polynomials :

InfProd_2884.png

InfProd_2885.png

InfProd_2886.png

InfProd_2887.png

InfProd_2888.png

InfProd_2889.png

Series of Zeta, PolyGamma, PolyLog and related :   ( Back to Top )

InfProd_2890.png

InfProd_2891.png

InfProd_2892.png

InfProd_2893.png

InfProd_2894.png

InfProd_2895.png

InfProd_2896.png

InfProd_2897.png

InfProd_2898.png

InfProd_2899.png

InfProd_2900.png

InfProd_2901.png

InfProd_2902.png

InfProd_2903.png

InfProd_2904.png

InfProd_2905.png

InfProd_2906.png

InfProd_2907.png

InfProd_2908.png

InfProd_2909.png

InfProd_2910.png

InfProd_2911.png

InfProd_2912.png

InfProd_2913.png

InfProd_2914.png

InfProd_2915.png

InfProd_2916.png

InfProd_2917.png

InfProd_2918.png

InfProd_2919.png

InfProd_2920.png

InfProd_2921.png

InfProd_2922.png

InfProd_2923.png

InfProd_2924.png

InfProd_2925.png

InfProd_2926.png

InfProd_2927.png

The next expression is an asymptotic approximation in s (better than 1 % ):

InfProd_2928.png

InfProd_2929.png

InfProd_2930.png

InfProd_2931.png

InfProd_2932.png

InfProd_2933.png

InfProd_2934.png

InfProd_2935.png

InfProd_2936.png

InfProd_2937.png

InfProd_2938.png

InfProd_2939.png

InfProd_2940.png

InfProd_2941.png

InfProd_2942.png

InfProd_2943.png

InfProd_2944.gif

Special Values of Zeta :

InfProd_2945.png

InfProd_2946.png

InfProd_2947.png

InfProd_2948.png

InfProd_2949.gif

InfProd_2950.png

InfProd_2951.png

InfProd_2952.png

InfProd_2953.png

PolyGamma :

InfProd_2954.png

InfProd_2955.png

InfProd_2956.png

InfProd_2957.png

InfProd_2958.png

InfProd_2959.png

InfProd_2960.png

InfProd_2961.png

InfProd_2962.png

InfProd_2963.png

InfProd_2964.png

InfProd_2965.png

InfProd_2966.png

InfProd_2967.png

InfProd_2968.png

InfProd_2969.png

InfProd_2970.png

InfProd_2971.png

InfProd_2972.png

InfProd_2973.png

InfProd_2974.png

InfProd_2975.png

InfProd_2976.png

InfProd_2977.png

InfProd_2978.png

InfProd_2979.png

InfProd_2980.png

InfProd_2981.png

InfProd_2982.png

InfProd_2983.png

InfProd_2984.png

InfProd_2985.png

InfProd_2986.gif

InfProd_2987.png

InfProd_2988.png

PolyLog and  LerchPhi :

InfProd_2989.png

InfProd_2990.png

InfProd_2991.png

InfProd_2992.png

InfProd_2993.png

InfProd_2994.png

InfProd_2995.png

InfProd_2996.png

InfProd_2997.png

InfProd_2998.png

InfProd_2999.png

InfProd_3000.png

InfProd_3001.png

The sum inside the large brackets above gives the Eulerian numbers .

InfProd_3002.gif

InfProd_3003.png

InfProd_3004.png

InfProd_3005.png

InfProd_3006.png

InfProd_3007.png

InfProd_3008.gif

InfProd_3009.png

InfProd_3010.png

InfProd_3011.png

InfProd_3012.png

InfProd_3013.png

InfProd_3014.png

InfProd_3015.png

InfProd_3016.png

InfProd_3017.png

InfProd_3018.png

InfProd_3019.png

InfProd_3020.png

InfProd_3021.png

InfProd_3022.png

InfProd_3023.png

InfProd_3024.png

InfProd_3025.png

InfProd_3026.png

InfProd_3027.png

InfProd_3028.png

InfProd_3029.png

InfProd_3030.png

InfProd_3031.png

InfProd_3032.png

InfProd_3033.png

InfProd_3034.png

InfProd_3035.png

InfProd_3036.png

InfProd_3037.png

InfProd_3038.png

InfProd_3039.png

InfProd_3040.png

InfProd_3041.png

InfProd_3042.png

InfProd_3043.png

InfProd_3044.png

InfProd_3045.png

InfProd_3046.png

InfProd_3047.png

InfProd_3048.png

From Reynolds’ LerchPhi equation (4.1) in https://arxiv.org/pdf/2306.12565.pdf :

InfProd_3049.png

InfProd_3050.png

InfProd_3051.png

InfProd_3052.png

Special Values of PolyLog and LerchPhi :

InfProd_3053.png

InfProd_3054.png

InfProd_3055.png

InfProd_3056.png

InfProd_3057.png

InfProd_3058.png

InfProd_3059.png

InfProd_3060.png

InfProd_3061.png

InfProd_3062.png

InfProd_3063.png

InfProd_3064.png

InfProd_3065.png

InfProd_3066.png

InfProd_3067.png

InfProd_3068.gif

InfProd_3069.png

InfProd_3070.gif

InfProd_3071.png

InfProd_3072.png

InfProd_3073.png

InfProd_3074.png

InfProd_3075.png

Series of Beta Functions :   ( Back to Top )

Recurrence relation : Beta[x, a + 1, b] + Beta[x, a, b + 1] = Beta[x, a, b];

InfProd_3076.png

InfProd_3077.png

InfProd_3078.png

InfProd_3079.png

InfProd_3080.png

InfProd_3081.png

InfProd_3082.png

InfProd_3083.png

InfProd_3084.png

InfProd_3085.png

InfProd_3086.png

InfProd_3087.png

InfProd_3088.png

InfProd_3089.png

InfProd_3090.png

InfProd_3091.png

InfProd_3092.png

InfProd_3093.png

InfProd_3094.png

InfProd_3095.png

InfProd_3096.png

InfProd_3097.png

InfProd_3098.png

InfProd_3099.png

InfProd_3100.png

InfProd_3101.png

InfProd_3102.png

InfProd_3103.png

InfProd_3104.png

InfProd_3105.png

InfProd_3106.png

InfProd_3107.png

InfProd_3108.png

InfProd_3109.png

InfProd_3110.png

InfProd_3111.png

InfProd_3112.png

InfProd_3113.png

InfProd_3114.png

InfProd_3115.png

Special  values  of  Beta related functions :

InfProd_3116.png

InfProd_3117.png

Series of Gamma Functions :   ( Back to Top )

InfProd_3118.png

Dougall' s Formula :

InfProd_3119.png

InfProd_3120.png

InfProd_3121.png

InfProd_3122.png

InfProd_3123.png

InfProd_3124.gif

InfProd_3125.png

InfProd_3126.png

InfProd_3127.png

InfProd_3128.png

InfProd_3129.png

InfProd_3130.png

InfProd_3131.png

InfProd_3132.png

InfProd_3133.png

InfProd_3134.png

InfProd_3135.png

InfProd_3136.png

InfProd_3137.png

InfProd_3138.png

InfProd_3139.png

InfProd_3140.png

InfProd_3141.png

note the offset of 1/2 that appears in the result of the second series above if a is set to 1.

InfProd_3142.png

InfProd_3143.png

InfProd_3144.png

InfProd_3145.png

InfProd_3146.png

InfProd_3147.png

InfProd_3148.png

The real part of the following Gamma series doesn' t converge :

InfProd_3149.png

InfProd_3150.png

InfProd_3151.png

InfProd_3152.png

InfProd_3153.png

InfProd_3154.png

InfProd_3155.png

Gamma Identities :

InfProd_3156.png

InfProd_3157.png

Special value of Gamma :

InfProd_3158.png

Special values of InverseGammaRegularized :

InfProd_3159.png

Series involving HarmonicNumber : ( Back To Top )

InfProd_3160.png

InfProd_3161.png

InfProd_3162.png

InfProd_3163.png

InfProd_3164.png

InfProd_3165.png

InfProd_3166.png

InfProd_3167.png

InfProd_3168.png

InfProd_3169.png

InfProd_3170.png

InfProd_3171.png

InfProd_3172.png

InfProd_3173.png

InfProd_3174.png

InfProd_3175.png

InfProd_3176.png

InfProd_3177.png

InfProd_3178.png

InfProd_3179.png

InfProd_3180.png

InfProd_3181.png

InfProd_3182.png

InfProd_3183.png

InfProd_3184.png

For n = 1 to 10    InfProd_3185.png is:

1: InfProd_3186.png -0.58224053
2: InfProd_3187.png -0.90797054
3: InfProd_3188.png -1.13055188
4: InfProd_3189.png -1.29927612
5: InfProd_3190.png -1.43505814
6: InfProd_3191.png -1.54863772
7: InfProd_3192.png -1.64624639
8: InfProd_3193.png -1.73181782
9: InfProd_3194.png -1.80799286
10: InfProd_3195.png -1.87662974

InfProd_3196.png

InfProd_3197.png

For n = 1 to 4     InfProd_3198.png is:

1: InfProd_3199.png 2.40411381
2: InfProd_3200.png 3.30565648
3: InfProd_3201.png 3.88459579
4: InfProd_3202.png 4.31204500

InfProd_3203.png

InfProd_3204.png

For n = 1 to 3     InfProd_3205.png is:

1: InfProd_3206.png -0.7512856
2: InfProd_3207.png -1.1496340
3: InfProd_3208.png -1.4185815

InfProd_3209.png

InfProd_3210.png

InfProd_3211.png

InfProd_3212.png

InfProd_3213.png

InfProd_3214.png

InfProd_3215.png

InfProd_3216.png

InfProd_3217.png

InfProd_3218.png

InfProd_3219.png

InfProd_3220.png

InfProd_3221.png

InfProd_3222.png

InfProd_3223.png

InfProd_3224.png

InfProd_3225.png

InfProd_3226.png

InfProd_3227.png

InfProd_3228.png

InfProd_3229.png

The notation in the Exponent of the hypergeometric function says : take the derivative with respect to the seventh argument and then give it the value 1 - a or 1 + a.

Series involving Hypergeometric Functions : ( Back to Top )

InfProd_3230.png

InfProd_3231.png

InfProd_3232.png

InfProd_3233.png

InfProd_3234.png

InfProd_3235.png

InfProd_3236.png

InfProd_3237.png

InfProd_3238.png

InfProd_3239.png

InfProd_3240.png

InfProd_3241.png

InfProd_3242.png

InfProd_3243.png

InfProd_3244.png

InfProd_3245.png

InfProd_3246.png

InfProd_3247.png

InfProd_3248.png

InfProd_3249.png

InfProd_3250.png

InfProd_3251.png

InfProd_3252.png

InfProd_3253.png

InfProd_3254.png

InfProd_3255.png

InfProd_3256.png

InfProd_3257.png

InfProd_3258.png

InfProd_3259.png

InfProd_3260.png

InfProd_3261.png

InfProd_3262.png

InfProd_3263.png

InfProd_3264.png

InfProd_3265.png

InfProd_3266.png

InfProd_3267.png

InfProd_3268.png

InfProd_3269.png

InfProd_3270.png

InfProd_3271.png

InfProd_3272.png

InfProd_3273.png

InfProd_3274.png

InfProd_3275.png

InfProd_3276.png

InfProd_3277.png

InfProd_3278.png

InfProd_3279.png

InfProd_3280.png

InfProd_3281.png

InfProd_3282.png

InfProd_3283.png

InfProd_3284.png

InfProd_3285.png

InfProd_3286.png

InfProd_3287.png

InfProd_3288.png

Special value:

InfProd_3289.png

Some Limits :   ( Back to Top )

InfProd_3290.png

InfProd_3291.png

InfProd_3292.gif

InfProd_3293.png

InfProd_3294.png

InfProd_3295.png

InfProd_3296.png

InfProd_3297.png

InfProd_3298.png

InfProd_3299.png

InfProd_3300.png

InfProd_3301.png

InfProd_3302.png

InfProd_3303.png

InfProd_3304.png

InfProd_3305.png

InfProd_3306.png

InfProd_3307.png

InfProd_3308.png

InfProd_3309.png

InfProd_3310.png

A few Integrals :   ( Back to Top )

InfProd_3311.png

Substitute  InfProd_3312.png   and the Feynman - Hibbs Integral

InfProd_3313.png

InfProd_3314.png

and derivatives :

InfProd_3315.png is the mth derivative with respect to a :

InfProd_3316.png

InfProd_3317.png

InfProd_3318.png

InfProd_3319.png

Iterated Expressions  ( Tetration ) :   ( Back to Top )

InfProd_3320.gif

InfProd_3321.png

InfProd_3322.gif

InfProd_3323.png

InfProd_3324.png

InfProd_3325.png

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: InfProd_3326.png
f[x] is not defined beyond the maximum of its inverse function InfProd_3327.png, namely  InfProd_3328.png< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < InfProd_3329.png as well, where it shows a kind of bifurcation.

Solve special recursions using a corresponding differential equation:

Consider recursions of the form f[k+1] = f[0] g[ f[k] ] + f[k] that converge to finite values for large k. They may sometimes be solved by using a differential equation.
If  f[0] ~ x/n, where some large k is  k ≤ n, then f[0] will be small and the differential equation to try to solve reads  y'[x] == g[ y[x] ] with boundary condition y[0] = x/n.
The recursion f[n] calculated starting from a certain value of x/n will then converge to y[x].
Several cases of g[ f[k] ] are presented.  All result in strictly monotonically increasing y[x] over their respective domain.
Verify that your selection of {a,b,c} and size of n is working out by numerical and graphical check,
(The value of n may have to be inceased in some cases for better convergence; if the limit n→ ∞ leads to y = 0, keep n at a large but finite value as shown in the Sin[] and Tan[] example below):

InfProd_3330.png

The points of the recursion are iteratively calculated and can be subsequently plotted together with the result in this case like

InfProd_3331.gif

You may have to adapt the Plot ranges and intervalls of x to the next exemplary cases :

InfProd_3332.png

InfProd_3333.png

InfProd_3334.png

InfProd_3335.png

InfProd_3336.png

InfProd_3337.png

InfProd_3338.png

InfProd_3339.png

InfProd_3340.png

InfProd_3341.png

InfProd_3342.png

InfProd_3343.png

InfProd_3344.png

InfProd_3345.png

InfProd_3346.png

InfProd_3347.png

InfProd_3348.png

InfProd_3349.png

InfProd_3350.png

InfProd_3351.png

Some Properties of ProductLog, LerchPhi and PolyLog   ( Back to Top )

For 1/e ≤ x    is ProductLog[ x Log[ x ]]        =    Log[ x ] .
For 0 ≤ x ≤ e    is ProductLog[ - Log[ x ] / x]    = - Log[ x ] .
For 0 ≤ x        is Log[ ProductLog[ x ] ]        =   Log[ x ] - ProductLog[ x ] .

InfProd_3352.png

InfProd_3353.png

InfProd_3354.png

InfProd_3355.png

1
1 2
1 8 6
1 22 58 24

InfProd_3356.png

For purely imaginary arguments (x ∈ R) the complex decomposition of LerchPhi is :

InfProd_3357.png

These carry over with a = 0 to PolyLog :

InfProd_3358.png

InfProd_3359.png

The imaginary part of LerchPhi[x, s, a] with 1 ≤ x ∈ R is given by :

InfProd_3360.png

And with a = 0 follows the imaginary part of PolyLog[ s, x] :

InfProd_3361.png

The complex decomposition of  InfProd_3362.png with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression :

InfProd_3363.png

explicitly for low s and b = 2 :

InfProd_3364.png InfProd_3365.png
InfProd_3366.png InfProd_3367.png
InfProd_3368.png InfProd_3369.png
InfProd_3370.png InfProd_3371.png
InfProd_3372.png InfProd_3373.png
InfProd_3374.png InfProd_3375.png

For all z ∈ C not on the real axis in ( -∞ < z < 1) and 0 ≤ {b, s} ∈ N the following inversion identity holds
(the If statement inserts a '+' in case of an imaginary part of z larger than zero, a '-' in all other cases) :

InfProd_3376.png

The real part of  InfProd_3377.pngwith 1 ≤ x ∈ R is also given by

InfProd_3378.png

For (b ∈ N) is

InfProd_3379.png

The real and imaginary parts of LerchPhi[ InfProd_3380.png, 2, 1/2 ] (on the unit circle) are

InfProd_3381.png

With Clausen type functions for LerchPhi defined as

InfProd_3382.png

InfProd_3383.png

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π, the even CLi and the odd SLi are expressible through Euler Polynomials),
the real and imaginary parts of InfProd_3384.pngInfProd_3385.png (on the unit circle) are

InfProd_3386.png

the expressions for InfProd_3387.png with lowest s being

InfProd_3388.png InfProd_3389.png
InfProd_3390.png InfProd_3391.png
InfProd_3392.png InfProd_3393.png
InfProd_3394.png InfProd_3395.png
InfProd_3396.png InfProd_3397.png
InfProd_3398.png InfProd_3399.png

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s.
The first non polynomial partnerfunctions are found to be

InfProd_3400.png

The function InfProd_3401.png has an interesting derivative :

InfProd_3402.png

that means the lower CLi and SLi are essentially derivatives of the higher ones.

With the LerchPhi index n being a negative Integer the function appears as a rational function :

InfProd_3403.png

With the PolyLog index being a negative Integer the function appears as a rational function :

InfProd_3404.png

With Clausen type functions defined as

InfProd_3405.png

InfProd_3406.png

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π,  the even Ci and the odd Si are expressible through Bernoulli Polynomials),
the real and imaginary parts of InfProd_3407.png (on the unit circle) are

InfProd_3408.png

the expressions for InfProd_3409.png with lowest s being

InfProd_3410.png

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s :

InfProd_3411.png

As before the derivative InfProd_3412.pngInfProd_3413.pngis InfProd_3414.png with lowered index.
The first non polynomial partnerfunctions are found to be

InfProd_3415.png

The complex decomposition of  PolyLog[s, x] with 1 ≤ x ∈ R and 0 ≤ s ∈ N can be obtained by the following expression :

InfProd_3416.png

explicitly for low s :

InfProd_3417.png InfProd_3418.png
InfProd_3419.png InfProd_3420.png
InfProd_3421.png InfProd_3422.png
InfProd_3423.png InfProd_3424.png
InfProd_3425.png InfProd_3426.png
InfProd_3427.png InfProd_3428.png
InfProd_3429.png InfProd_3430.png

For real x < 1  is :

InfProd_3431.png

InfProd_3432.png Log[x]-Log[1-x]-i π
InfProd_3433.png InfProd_3434.png
InfProd_3435.png InfProd_3436.png
InfProd_3437.png InfProd_3438.png
InfProd_3439.png InfProd_3440.png
InfProd_3441.png InfProd_3442.png

For all z ∈ C and not on the real axis in ( 0 ≤ z < 1 ) and 0 ≤  s ∈ N the following inversion identity holds :

InfProd_3443.png

LerchPhi and PolyLog display a similar (alternating with s) scheme in their real and imaginary parts :

InfProd_3444.png

InfProd_3445.png

InfProd_3446.png

InfProd_3447.png

InfProd_3448.png

InfProd_3449.png

InfProd_3450.png

InfProd_3451.png

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
InfProd_3452.png InfProd_3453.png InfProd_3454.png
InfProd_3455.png InfProd_3456.png InfProd_3457.png
InfProd_3458.png InfProd_3459.png InfProd_3460.png
InfProd_3461.png InfProd_3462.png InfProd_3463.png
InfProd_3464.png InfProd_3465.png InfProd_3466.png
InfProd_3467.png InfProd_3468.png InfProd_3469.png

They are symmetric or antisymmetric (depending on n) with respect to x = 1/2 :

InfProd_3470.png

Connection to Bernoulli and Euler numbers :

InfProd_3471.png

Clausen functions and integral :

InfProd_3472.png

InfProd_3473.png

InfProd_3474.png

InfProd_3475.png

Contributors :

(R1) Udo Ausserlechner, Infineon, per email

(R2) Professor Don Zagier, MPI für Mathematik, Bonn

(R3) Stephen, per email

(R4) Erik Vigren, IRF, Uppsala, per email

InfProd_3476.png

InfProd_3477.gif InfProd_3478.gif
InfProd_3479.gif InfProd_3480.gif

InfProd_3481.png

InfProd_3482.gif InfProd_3483.gif
InfProd_3484.gif InfProd_3485.gif

InfProd_3486.png

InfProd_3487.png

InfProd_3488.gif

InfProd_3489.png

InfProd_3490.gif

InfProd_3491.png

InfProd_3492.gif

InfProd_3493.png

InfProd_3494.gif

InfProd_3495.png

InfProd_3496.gif

InfProd_3497.png

InfProd_3498.png InfProd_3499.png
InfProd_3500.png InfProd_3501.png
InfProd_3502.png InfProd_3503.png
InfProd_3504.png InfProd_3505.png
InfProd_3506.png InfProd_3507.png

InfProd_3508.png

-Log[1-x] InfProd_3509.png
PolyLog[2,x] InfProd_3510.png
PolyLog[3,x] InfProd_3511.png
PolyLog[4,x] InfProd_3512.png
PolyLog[5,x] InfProd_3513.png

InfProd_3514.png

InfProd_3515.gif

InfProd_3516.gif

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InfProd_3561.gif

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InfProd_3594.gif

InfProd_3595.png

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InfProd_3597.png

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InfProd_3599.png

InfProd_3600.png

InfProd_3601.png

Jacobi:

InfProd_3602.png

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InfProd_3824.png

InfProd_3825.png

InfProd_3826.png

InfProd_3827.png

InfProd_3828.png

InfProd_3829.png

InfProd_3830.png

InfProd_3831.png

InfProd_3832.png

InfProd_3833.png

InfProd_3834.png

InfProd_3835.png

InfProd_3836.png

InfProd_3837.png

InfProd_3838.png

InfProd_3839.png

InfProd_3840.png

InfProd_3841.png

InfProd_3842.png

InfProd_3843.png

InfProd_3844.png

InfProd_3845.png

InfProd_3846.png

InfProd_3847.png

InfProd_3848.png

InfProd_3849.png

InfProd_3850.png

InfProd_3851.png

InfProd_3852.png

InfProd_3853.png

InfProd_3854.png

InfProd_3855.png

InfProd_3856.png

InfProd_3857.png

InfProd_3858.png

InfProd_3859.png

InfProd_3860.png

InfProd_3861.png

InfProd_3862.png

InfProd_3863.png

InfProd_3864.png

InfProd_3865.png

InfProd_3866.png

InfProd_3867.png

InfProd_3868.png

InfProd_3869.png

InfProd_3870.png

InfProd_3871.png

InfProd_3872.png

InfProd_3873.png

InfProd_3874.png

InfProd_3875.png

InfProd_3876.png

InfProd_3877.png

InfProd_3878.png

InfProd_3879.png

InfProd_3880.png

InfProd_3881.png

InfProd_3882.png

InfProd_3883.png

InfProd_3884.png

InfProd_3885.png

InfProd_3886.png

InfProd_3887.png

InfProd_3888.png

InfProd_3889.png

InfProd_3890.png

InfProd_3891.png

InfProd_3892.png

InfProd_3893.png

InfProd_3894.png

InfProd_3895.png

InfProd_3896.png

InfProd_3897.png

InfProd_3898.png

InfProd_3899.png

InfProd_3900.png

InfProd_3901.png

InfProd_3902.png

InfProd_3903.png

InfProd_3904.png

InfProd_3905.png

InfProd_3906.png

InfProd_3907.png

InfProd_3908.png

InfProd_3909.png

InfProd_3910.gif

InfProd_3911.png

InfProd_3912.png

InfProd_3913.png

InfProd_3914.png

InfProd_3915.png

InfProd_3916.png

InfProd_3917.png

InfProd_3918.png

InfProd_3919.gif

InfProd_3920.png

InfProd_3921.png

InfProd_3922.png

InfProd_3923.png

InfProd_3924.png

InfProd_3925.png

InfProd_3926.png

InfProd_3927.png

InfProd_3928.gif

InfProd_3929.png

InfProd_3930.png

InfProd_3931.png

InfProd_3932.png

InfProd_3933.png

InfProd_3934.png

InfProd_3935.png

InfProd_3936.png

InfProd_3937.png

InfProd_3938.png

InfProd_3939.png

InfProd_3940.png

InfProd_3941.png

InfProd_3942.png

InfProd_3943.png

InfProd_3944.png

InfProd_3945.png

InfProd_3946.png

InfProd_3947.png

InfProd_3948.png

InfProd_3949.png

InfProd_3950.png

InfProd_3951.png

InfProd_3952.png

InfProd_3953.png

InfProd_3954.png

InfProd_3955.png

InfProd_3956.png

InfProd_3957.png

InfProd_3958.png

InfProd_3959.png

InfProd_3960.png

InfProd_3961.png

InfProd_3962.png

InfProd_3963.png

InfProd_3964.png

InfProd_3965.png

InfProd_3966.png

InfProd_3967.png

InfProd_3968.png

InfProd_3969.png

InfProd_3970.png

InfProd_3971.png

InfProd_3972.png

InfProd_3973.png

InfProd_3974.png

InfProd_3975.png

InfProd_3976.png

InfProd_3977.png

InfProd_3978.png

InfProd_3979.png

InfProd_3980.png

InfProd_3981.png

InfProd_3982.png

InfProd_3983.png

InfProd_3984.png

InfProd_3985.png

InfProd_3986.gif

InfProd_3987.png

InfProd_3988.png

InfProd_3989.png

InfProd_3990.png

InfProd_3991.png

InfProd_3992.png

InfProd_3993.png

InfProd_3994.png

InfProd_3995.png

InfProd_3996.png

InfProd_3997.png

InfProd_3998.png

InfProd_3999.png

InfProd_4000.png

InfProd_4001.png

InfProd_4002.png

InfProd_4003.png

InfProd_4004.png

InfProd_4005.png

InfProd_4006.png

InfProd_4007.png

InfProd_4008.png

InfProd_4009.png

InfProd_4010.png

InfProd_4011.png

InfProd_4012.png

InfProd_4013.png

InfProd_4014.png

InfProd_4015.png

InfProd_4016.png

InfProd_4017.png

InfProd_4018.png

InfProd_4019.png

InfProd_4020.png

InfProd_4021.png

InfProd_4022.png

InfProd_4023.png

InfProd_4024.png

InfProd_4025.png

InfProd_4026.png

InfProd_4027.png

InfProd_4028.png

InfProd_4029.png

InfProd_4030.png

InfProd_4031.png

InfProd_4032.png

InfProd_4033.png

InfProd_4034.png

InfProd_4035.png

InfProd_4036.png

InfProd_4037.png

InfProd_4038.png

InfProd_4039.png

InfProd_4040.png

InfProd_4041.png

InfProd_4042.png

InfProd_4043.png

InfProd_4044.png

InfProd_4045.png

InfProd_4046.png

InfProd_4047.png

InfProd_4048.png

InfProd_4049.png

InfProd_4050.png

InfProd_4051.png

InfProd_4052.png

InfProd_4053.png

InfProd_4054.png

InfProd_4055.png

InfProd_4056.png

InfProd_4057.png

InfProd_4058.png

InfProd_4059.png

InfProd_4060.png

InfProd_4061.png

InfProd_4062.png

InfProd_4063.png

InfProd_4064.png

InfProd_4065.png

InfProd_4066.png

InfProd_4067.png

InfProd_4068.png

InfProd_4069.png

InfProd_4070.png

InfProd_4071.png

InfProd_4072.png

InfProd_4073.png

InfProd_4074.png

InfProd_4075.png

InfProd_4076.png

InfProd_4077.png

InfProd_4078.png

InfProd_4079.png

InfProd_4080.png

InfProd_4081.png

InfProd_4082.png

InfProd_4083.gif

InfProd_4084.png

InfProd_4085.png

InfProd_4086.png

InfProd_4087.png

InfProd_4088.png

InfProd_4089.png

InfProd_4090.png

InfProd_4091.png

InfProd_4092.png

InfProd_4093.png

InfProd_4094.png

InfProd_4095.png

InfProd_4096.png

InfProd_4097.png

InfProd_4098.png

InfProd_4099.png

InfProd_4100.png

InfProd_4101.png

InfProd_4102.png

InfProd_4103.png

InfProd_4104.png

InfProd_4105.png

InfProd_4106.png

InfProd_4107.png

InfProd_4108.png

InfProd_4109.png

InfProd_4110.png

InfProd_4111.png

InfProd_4112.png

InfProd_4113.png

InfProd_4114.png

InfProd_4115.png

InfProd_4116.png

InfProd_4117.png

InfProd_4118.png

InfProd_4119.png

InfProd_4120.png

InfProd_4121.png

InfProd_4122.png

InfProd_4123.png

InfProd_4124.png

InfProd_4125.png

InfProd_4126.png

InfProd_4127.png

InfProd_4128.png

InfProd_4129.png

InfProd_4130.png

InfProd_4131.gif

InfProd_4132.png

InfProd_4133.png

InfProd_4134.png

InfProd_4135.gif

InfProd_4136.png

InfProd_4137.png

InfProd_4138.png

InfProd_4139.png

InfProd_4140.png

InfProd_4141.png

InfProd_4142.png

InfProd_4143.png

InfProd_4144.png

InfProd_4145.png

InfProd_4146.png

InfProd_4147.png

InfProd_4148.png

InfProd_4149.png

InfProd_4150.png

InfProd_4151.png

InfProd_4152.png

InfProd_4153.png

InfProd_4154.png

InfProd_4155.png

InfProd_4156.png

InfProd_4157.png

InfProd_4158.png

InfProd_4159.png

InfProd_4160.png

InfProd_4161.png

InfProd_4162.png

InfProd_4163.png

InfProd_4164.png

InfProd_4165.png

InfProd_4166.png

InfProd_4167.png

InfProd_4168.png

InfProd_4169.png

InfProd_4170.png

InfProd_4171.gif

InfProd_4172.png

InfProd_4173.png

InfProd_4174.gif

InfProd_4175.png

InfProd_4176.png

InfProd_4177.gif

InfProd_4178.png

InfProd_4179.png

InfProd_4180.gif

InfProd_4181.png

InfProd_4182.png

InfProd_4183.gif

InfProd_4184.png

InfProd_4185.png

InfProd_4186.gif

InfProd_4187.png

InfProd_4188.png

InfProd_4189.png

InfProd_4190.png

InfProd_4191.png

InfProd_4192.png

InfProd_4193.png

InfProd_4194.png

InfProd_4195.png

InfProd_4196.png

InfProd_4197.png

InfProd_4198.png

InfProd_4199.png

InfProd_4200.png

InfProd_4201.png

InfProd_4202.png

InfProd_4203.png

InfProd_4204.png

InfProd_4205.png

InfProd_4206.png

InfProd_4207.png

InfProd_4208.png

InfProd_4209.png

InfProd_4210.png

InfProd_4211.gif

InfProd_4212.png

InfProd_4213.png

InfProd_4214.png

InfProd_4215.png

InfProd_4216.png

InfProd_4217.png

InfProd_4218.png

InfProd_4219.png

InfProd_4220.png

InfProd_4221.png

InfProd_4222.png

InfProd_4223.png

InfProd_4224.png

InfProd_4225.png

InfProd_4226.png

InfProd_4227.png

InfProd_4228.png

InfProd_4229.png

InfProd_4230.png

InfProd_4231.png

InfProd_4232.png

InfProd_4233.png

InfProd_4234.png

InfProd_4235.png

InfProd_4236.png

InfProd_4237.png

InfProd_4238.png

InfProd_4239.png

InfProd_4240.png

InfProd_4241.png

InfProd_4242.png

InfProd_4243.png

InfProd_4244.png

InfProd_4245.png

InfProd_4246.png

InfProd_4247.png

InfProd_4248.png

InfProd_4249.png

InfProd_4250.png

InfProd_4251.png

InfProd_4252.png

InfProd_4253.png

InfProd_4254.png

InfProd_4255.png

InfProd_4256.png

InfProd_4257.gif

InfProd_4258.png

InfProd_4259.png

InfProd_4260.png

InfProd_4261.gif

InfProd_4262.png

InfProd_4263.png

InfProd_4264.png

λ→0

InfProd_4265.png

InfProd_4266.png

InfProd_4267.png

InfProd_4268.png

InfProd_4269.png

InfProd_4270.png

InfProd_4271.png

InfProd_4272.png

InfProd_4273.png

InfProd_4274.png

InfProd_4275.png

InfProd_4276.png

InfProd_4277.png

InfProd_4278.png

InfProd_4279.png

InfProd_4280.png

InfProd_4281.png

InfProd_4282.png

InfProd_4283.png

InfProd_4284.png

InfProd_4285.png

InfProd_4286.png

InfProd_4287.png

InfProd_4288.png

InfProd_4289.png

InfProd_4290.png

InfProd_4291.gif

InfProd_4292.png

InfProd_4293.png

InfProd_4294.png

InfProd_4295.png

InfProd_4296.png

InfProd_4297.png

InfProd_4298.png

InfProd_4299.png

InfProd_4300.png

InfProd_4301.png

InfProd_4302.png

InfProd_4303.png

InfProd_4304.png

InfProd_4305.png

InfProd_4306.png

InfProd_4307.png

InfProd_4308.png

InfProd_4309.png

InfProd_4310.png

InfProd_4311.png

InfProd_4312.png

InfProd_4313.png

InfProd_4314.png

InfProd_4315.png

InfProd_4316.png

InfProd_4317.png

InfProd_4318.png

Sinh:

InfProd_4319.png

InfProd_4320.png

InfProd_4321.png

InfProd_4322.png

InfProd_4323.png

InfProd_4324.png

InfProd_4325.png

InfProd_4326.png

InfProd_4327.png

InfProd_4328.png

InfProd_4329.png

InfProd_4330.png

Sin:
InfProd_4331.pngis from the number of n negative factors in the denominator…

InfProd_4332.png

InfProd_4333.png

InfProd_4334.png

InfProd_4335.png

InfProd_4336.png

InfProd_4337.png

InfProd_4338.png

InfProd_4339.png

Cosh:

InfProd_4340.png

InfProd_4341.png

InfProd_4342.png

InfProd_4343.png

Cos:

InfProd_4344.png

InfProd_4345.png

InfProd_4346.png

InfProd_4347.png

Tan:

InfProd_4348.png

InfProd_4349.png

InfProd_4350.png

InfProd_4351.png

Tanh:

InfProd_4352.png

InfProd_4353.png

InfProd_4354.png

InfProd_4355.png

InfProd_4356.png

InfProd_4357.png

InfProd_4358.png

InfProd_4359.png

InfProd_4360.png

InfProd_4361.png

InfProd_4362.png

InfProd_4363.png

InfProd_4364.png

InfProd_4365.png

InfProd_4366.gif

InfProd_4367.png

InfProd_4368.png

InfProd_4369.png

InfProd_4370.png

InfProd_4371.png

InfProd_4372.png

InfProd_4373.png

InfProd_4374.png

InfProd_4375.png

InfProd_4376.png

InfProd_4377.png

InfProd_4378.png

InfProd_4379.png

InfProd_4380.png

InfProd_4381.png

InfProd_4382.png

InfProd_4383.png

InfProd_4384.png

InfProd_4385.png

InfProd_4386.png

InfProd_4387.png

InfProd_4388.png

InfProd_4389.png

InfProd_4390.png

InfProd_4391.png

InfProd_4392.png

InfProd_4393.png

InfProd_4394.png

InfProd_4395.png

InfProd_4396.png

InfProd_4397.png

InfProd_4398.png

InfProd_4399.png

InfProd_4400.png

InfProd_4401.png

InfProd_4402.png

InfProd_4403.png

z→-z

InfProd_4404.png

InfProd_4405.png

InfProd_4406.png

InfProd_4407.png

InfProd_4408.png

InfProd_4409.png

InfProd_4410.png

InfProd_4411.png

InfProd_4412.png

InfProd_4413.png

InfProd_4414.png

InfProd_4415.png

InfProd_4416.png

InfProd_4417.png

InfProd_4418.png

InfProd_4419.png

InfProd_4420.png

InfProd_4421.png

InfProd_4422.png

InfProd_4423.png

InfProd_4424.png

InfProd_4425.png

InfProd_4426.png

InfProd_4427.png

InfProd_4428.png

InfProd_4429.png

InfProd_4430.png

InfProd_4431.png

InfProd_4432.png

InfProd_4433.png

InfProd_4434.png

InfProd_4435.png

InfProd_4436.png

InfProd_4437.png

InfProd_4438.png

InfProd_4439.png

InfProd_4440.png

InfProd_4441.png

InfProd_4442.png

InfProd_4443.png

InfProd_4444.gif

InfProd_4445.png

InfProd_4446.png

InfProd_4447.png

InfProd_4448.png

InfProd_4449.gif

InfProd_4450.png

InfProd_4451.png

InfProd_4452.png

InfProd_4453.png

InfProd_4454.gif

InfProd_4455.png

InfProd_4456.png

InfProd_4457.png

InfProd_4458.png

InfProd_4459.png

InfProd_4460.png

InfProd_4461.png

InfProd_4462.png

InfProd_4463.png

InfProd_4464.png

InfProd_4465.png

InfProd_4466.png

InfProd_4467.png

InfProd_4468.png

InfProd_4469.png

InfProd_4470.png

InfProd_4471.png

InfProd_4472.png

InfProd_4473.png

InfProd_4474.png

InfProd_4475.png

InfProd_4476.png

InfProd_4477.png

InfProd_4478.png

InfProd_4479.png

InfProd_4480.png

InfProd_4481.png

InfProd_4482.png

InfProd_4483.png

InfProd_4484.png

InfProd_4485.png

InfProd_4486.png

InfProd_4487.png

InfProd_4488.png

InfProd_4489.png

InfProd_4490.png

InfProd_4491.png

InfProd_4492.png

InfProd_4493.png

InfProd_4494.png

InfProd_4495.png

InfProd_4496.png

InfProd_4497.png

InfProd_4498.png

InfProd_4499.png

InfProd_4500.png

InfProd_4501.png

InfProd_4502.png

InfProd_4503.png

InfProd_4504.png

InfProd_4505.png

InfProd_4506.png

InfProd_4507.png

InfProd_4508.png

InfProd_4509.png

InfProd_4510.png

InfProd_4511.png

InfProd_4512.png

InfProd_4513.png

InfProd_4514.png

InfProd_4515.png

InfProd_4516.png

InfProd_4517.png

InfProd_4518.png

InfProd_4519.png

InfProd_4520.png

InfProd_4521.png

InfProd_4522.png

InfProd_4523.png

InfProd_4524.png

InfProd_4525.png

InfProd_4526.png

InfProd_4527.png

InfProd_4528.png

InfProd_4529.png

InfProd_4530.png

InfProd_4531.png

InfProd_4532.png

InfProd_4533.png

InfProd_4534.png

InfProd_4535.png

InfProd_4536.png

InfProd_4537.png

InfProd_4538.png

InfProd_4539.png

InfProd_4540.png

InfProd_4541.png

InfProd_4542.png

InfProd_4543.png

InfProd_4544.png

InfProd_4545.png

InfProd_4546.png

InfProd_4547.png

InfProd_4548.png

InfProd_4549.png

InfProd_4550.png

InfProd_4551.png

InfProd_4552.png

InfProd_4553.png

InfProd_4554.png

InfProd_4555.png

InfProd_4556.png

InfProd_4557.png

InfProd_4558.png

InfProd_4559.png

InfProd_4560.png

InfProd_4561.png

InfProd_4562.png

InfProd_4563.png

InfProd_4564.png

InfProd_4565.png

InfProd_4566.png

InfProd_4567.png

InfProd_4568.png

InfProd_4569.png

InfProd_4570.png

InfProd_4571.png

InfProd_4572.png

InfProd_4573.png

InfProd_4574.png

InfProd_4575.png

InfProd_4576.png

InfProd_4577.png

InfProd_4578.png

InfProd_4579.png

InfProd_4580.png

InfProd_4581.png

InfProd_4582.png

InfProd_4583.gif

InfProd_4584.png

InfProd_4585.png

InfProd_4586.png

InfProd_4587.gif

InfProd_4588.png

InfProd_4589.png

InfProd_4590.png

InfProd_4591.png

InfProd_4592.png

InfProd_4593.png

InfProd_4594.png

InfProd_4595.png

InfProd_4596.png

InfProd_4597.png

InfProd_4598.png

InfProd_4599.png

InfProd_4600.png

InfProd_4601.png

InfProd_4602.png

InfProd_4603.png

InfProd_4604.png

InfProd_4605.png

InfProd_4606.png

InfProd_4607.png

InfProd_4608.png

InfProd_4609.png

InfProd_4610.png

InfProd_4611.png

InfProd_4612.png

InfProd_4613.png

InfProd_4614.png

InfProd_4615.png

InfProd_4616.png

InfProd_4617.png

Re[z]<>k &&Im[z]<>k

InfProd_4618.png

InfProd_4619.png

InfProd_4620.png

InfProd_4621.png

InfProd_4622.png

InfProd_4623.png

InfProd_4624.png

InfProd_4625.png

InfProd_4626.png

InfProd_4627.png

InfProd_4628.png

InfProd_4629.png

InfProd_4630.png

InfProd_4631.png

InfProd_4632.png

InfProd_4633.png

InfProd_4634.png

InfProd_4635.png

InfProd_4636.png

InfProd_4637.gif

InfProd_4638.png

InfProd_4639.png

InfProd_4640.gif

InfProd_4641.png

InfProd_4642.png

InfProd_4643.png

InfProd_4644.png

InfProd_4645.png

InfProd_4646.png

InfProd_4647.png

InfProd_4648.png

InfProd_4649.png

InfProd_4650.png

InfProd_4651.png

InfProd_4652.png

InfProd_4653.png

InfProd_4654.png

InfProd_4655.png

InfProd_4656.png

InfProd_4657.png

InfProd_4658.png

InfProd_4659.png

InfProd_4660.png

InfProd_4661.png

InfProd_4662.png

InfProd_4663.png

InfProd_4664.gif

InfProd_4665.png

InfProd_4666.png

InfProd_4667.png

InfProd_4668.png

InfProd_4669.png

InfProd_4670.png

InfProd_4671.png

InfProd_4672.png

InfProd_4673.png

InfProd_4674.png

InfProd_4675.png

InfProd_4676.png

InfProd_4677.png

InfProd_4678.png

InfProd_4679.png

InfProd_4680.png

InfProd_4681.png

InfProd_4682.png

InfProd_4683.png

InfProd_4684.png

InfProd_4685.png

InfProd_4686.png

InfProd_4687.png

InfProd_4688.png

InfProd_4689.png

InfProd_4690.png

InfProd_4691.png

InfProd_4692.png

InfProd_4693.png

InfProd_4694.png

InfProd_4695.png

InfProd_4696.png

InfProd_4697.png

InfProd_4698.png

InfProd_4699.png

InfProd_4700.png

InfProd_4701.png

InfProd_4702.png

InfProd_4703.png

InfProd_4704.png

InfProd_4705.png

InfProd_4706.png

InfProd_4707.png

InfProd_4708.png

InfProd_4709.png

InfProd_4710.png

InfProd_4711.png

InfProd_4712.png

InfProd_4713.png

InfProd_4714.png

InfProd_4715.png

InfProd_4716.png

InfProd_4717.png

InfProd_4718.png

InfProd_4719.png

InfProd_4720.png

InfProd_4721.png

InfProd_4722.png

InfProd_4723.png

InfProd_4724.png

InfProd_4725.png

InfProd_4726.png

InfProd_4727.png

InfProd_4728.png

InfProd_4729.png

InfProd_4730.png

InfProd_4731.gif

InfProd_4732.png

InfProd_4733.png

InfProd_4734.png

InfProd_4735.png

InfProd_4736.png

InfProd_4737.png

InfProd_4738.png

InfProd_4739.png

InfProd_4740.png

InfProd_4741.png

InfProd_4742.png

InfProd_4743.png

InfProd_4744.png

InfProd_4745.png

InfProd_4746.png

InfProd_4747.png

InfProd_4748.png

InfProd_4749.png

InfProd_4750.png

InfProd_4751.png

InfProd_4752.png

InfProd_4753.png

InfProd_4754.png

InfProd_4755.png

InfProd_4756.png

InfProd_4757.png

InfProd_4758.png

InfProd_4759.png

InfProd_4760.png

InfProd_4761.png

InfProd_4762.png

InfProd_4763.png

InfProd_4764.png

InfProd_4765.png

InfProd_4766.png

InfProd_4767.png

InfProd_4768.png

InfProd_4769.png

InfProd_4770.png

InfProd_4771.png

InfProd_4772.png

InfProd_4773.png

InfProd_4774.png

InfProd_4775.png

InfProd_4776.png

InfProd_4777.png

InfProd_4778.png

InfProd_4779.png

InfProd_4780.png

InfProd_4781.png

InfProd_4782.png

InfProd_4783.png

InfProd_4784.png

InfProd_4785.png

InfProd_4786.png

InfProd_4787.png

InfProd_4788.png

InfProd_4789.png

InfProd_4790.png

InfProd_4791.png

InfProd_4792.png

InfProd_4793.png

InfProd_4794.png

InfProd_4795.png

InfProd_4796.png

InfProd_4797.png

InfProd_4798.png

InfProd_4799.png

InfProd_4800.png

InfProd_4801.png

InfProd_4802.png

InfProd_4803.png

InfProd_4804.png

InfProd_4805.png

InfProd_4806.png

InfProd_4807.png

InfProd_4808.png

InfProd_4809.png

InfProd_4810.png

InfProd_4811.png

InfProd_4812.png

InfProd_4813.png

InfProd_4814.png

InfProd_4815.png

InfProd_4816.png

InfProd_4817.png

InfProd_4818.png

InfProd_4819.png

InfProd_4820.png

InfProd_4821.png

InfProd_4822.png

InfProd_4823.png

InfProd_4824.png

InfProd_4825.png

InfProd_4826.png

InfProd_4827.png

InfProd_4828.png

InfProd_4829.png

InfProd_4830.png

InfProd_4831.png

InfProd_4832.png

InfProd_4833.png

InfProd_4834.png

InfProd_4835.png

InfProd_4836.png

InfProd_4837.png

InfProd_4838.png

InfProd_4839.png

InfProd_4840.png

InfProd_4841.png

InfProd_4842.gif

InfProd_4843.png

Reihenentwicklung von m:

InfProd_4844.png

InfProd_4845.png

InfProd_4846.png

InfProd_4847.png

InfProd_4848.png

InfProd_4849.png

InfProd_4850.png

InfProd_4851.png

InfProd_4852.png

InfProd_4853.png

InfProd_4854.png

InfProd_4855.png

InfProd_4856.png

InfProd_4857.png

InfProd_4858.png

InfProd_4859.png

InfProd_4860.png

InfProd_4861.png

InfProd_4862.png

InfProd_4863.png

InfProd_4864.png

InfProd_4865.png

InfProd_4866.png

InfProd_4867.png

InfProd_4868.png

InfProd_4869.png

InfProd_4870.png

InfProd_4871.png

InfProd_4872.png

InfProd_4873.png

InfProd_4874.png

InfProd_4875.png

InfProd_4876.png

InfProd_4877.png

InfProd_4878.png

InfProd_4879.png

InfProd_4880.png

InfProd_4881.png

InfProd_4882.png

InfProd_4883.png

InfProd_4884.png

InfProd_4885.png

InfProd_4886.png

InfProd_4887.png

InfProd_4888.png

InfProd_4889.png

InfProd_4890.png

InfProd_4891.png

InfProd_4892.png

InfProd_4893.png

InfProd_4894.png

InfProd_4895.png

InfProd_4896.png

InfProd_4897.png

InfProd_4898.png

InfProd_4899.png

InfProd_4900.png

InfProd_4901.png

InfProd_4902.png

InfProd_4903.png

InfProd_4904.png

InfProd_4905.png

InfProd_4906.png

InfProd_4907.png

InfProd_4908.png

InfProd_4909.png

InfProd_4910.png

InfProd_4911.png

InfProd_4912.png

InfProd_4913.png

InfProd_4914.png

InfProd_4915.png

InfProd_4916.png

InfProd_4917.png

InfProd_4918.png

InfProd_4919.png

InfProd_4920.png

InfProd_4921.png

InfProd_4922.png

InfProd_4923.png

InfProd_4924.png

InfProd_4925.png

InfProd_4926.png

InfProd_4927.png

InfProd_4928.png

InfProd_4929.png

InfProd_4930.png

InfProd_4931.png

InfProd_4932.png

InfProd_4933.png

InfProd_4934.png

InfProd_4935.png

InfProd_4936.png

InfProd_4937.png

InfProd_4938.png

InfProd_4939.png

InfProd_4940.png

InfProd_4941.png

InfProd_4942.png

InfProd_4943.png

InfProd_4944.png

InfProd_4945.png

InfProd_4946.png

InfProd_4947.png

InfProd_4948.png

InfProd_4949.png

InfProd_4950.png

InfProd_4951.png

InfProd_4952.png

InfProd_4953.png

InfProd_4954.png

InfProd_4955.png

InfProd_4956.png

InfProd_4957.png

InfProd_4958.png

InfProd_4959.png

InfProd_4960.png

InfProd_4961.png

InfProd_4962.png

InfProd_4963.png

InfProd_4964.png

InfProd_4965.png

InfProd_4966.png

InfProd_4967.png

InfProd_4968.png

InfProd_4969.png

InfProd_4970.png

InfProd_4971.png

InfProd_4972.png

InfProd_4973.png

InfProd_4974.png

InfProd_4975.png

InfProd_4976.png

InfProd_4977.png

InfProd_4978.png

InfProd_4979.png

InfProd_4980.png

InfProd_4981.png

InfProd_4982.png

InfProd_4983.png

InfProd_4984.png

InfProd_4985.png

InfProd_4986.png

InfProd_4987.png

InfProd_4988.png

InfProd_4989.png

InfProd_4990.png

InfProd_4991.png

InfProd_4992.png

InfProd_4993.png

InfProd_4994.png

InfProd_4995.png

InfProd_4996.png

InfProd_4997.png

InfProd_4998.png

InfProd_4999.png

InfProd_5000.png

InfProd_5001.png

InfProd_5002.png

InfProd_5003.png

InfProd_5004.png

InfProd_5005.png

InfProd_5006.png

InfProd_5007.png

InfProd_5008.png

InfProd_5009.png

InfProd_5010.png

InfProd_5011.png

InfProd_5012.png

InfProd_5013.png

InfProd_5014.png

InfProd_5015.png

InfProd_5016.png

InfProd_5017.png

InfProd_5018.png

InfProd_5019.png

InfProd_5020.png

InfProd_5021.png

InfProd_5022.png

InfProd_5023.png

InfProd_5024.png

InfProd_5025.png

InfProd_5026.png

InfProd_5027.png

InfProd_5028.png

InfProd_5029.png

InfProd_5030.png

InfProd_5031.png

InfProd_5032.png

InfProd_5033.png

InfProd_5034.png

InfProd_5035.png

Glaisher:

InfProd_5036.png

InfProd_5037.png

InfProd_5038.png

InfProd_5039.png

InfProd_5040.png

InfProd_5041.png

InfProd_5042.png

InfProd_5043.png

InfProd_5044.png

InfProd_5045.png

InfProd_5046.png

InfProd_5047.png

InfProd_5048.png

InfProd_5049.png

InfProd_5050.png

InfProd_5051.png

InfProd_5052.png

InfProd_5053.png

InfProd_5054.png

InfProd_5055.png

InfProd_5056.png

InfProd_5057.png

InfProd_5058.png

InfProd_5059.png

InfProd_5060.png

InfProd_5061.png

InfProd_5062.png

InfProd_5063.png

InfProd_5064.png

InfProd_5065.png

InfProd_5066.png

InfProd_5067.png

InfProd_5068.png

InfProd_5069.png

InfProd_5070.png

InfProd_5071.png

InfProd_5072.png

InfProd_5073.png

InfProd_5074.png

InfProd_5075.png

InfProd_5076.png

InfProd_5077.png

InfProd_5078.png

InfProd_5079.png

InfProd_5080.png

InfProd_5081.png

InfProd_5082.png

InfProd_5083.png

InfProd_5084.png

InfProd_5085.png

InfProd_5086.png

InfProd_5087.png

InfProd_5088.png

InfProd_5089.png

InfProd_5090.png

InfProd_5091.png

InfProd_5092.png

InfProd_5093.png

InfProd_5094.png

InfProd_5095.png

InfProd_5096.png

InfProd_5097.png

InfProd_5098.png

InfProd_5099.png

InfProd_5100.png

InfProd_5101.png

InfProd_5102.png

InfProd_5103.png

InfProd_5104.png

InfProd_5105.png

InfProd_5106.png

InfProd_5107.png

InfProd_5108.png

InfProd_5109.png

InfProd_5110.png

InfProd_5111.png

InfProd_5112.png

InfProd_5113.png

InfProd_5114.png

InfProd_5115.png

InfProd_5116.png

InfProd_5117.png

InfProd_5118.png

InfProd_5119.png

InfProd_5120.png

InfProd_5121.png

InfProd_5122.png

InfProd_5123.png

InfProd_5124.png

InfProd_5125.png

InfProd_5126.png

InfProd_5127.png

InfProd_5128.png

InfProd_5129.png

InfProd_5130.png

InfProd_5131.png

InfProd_5132.png

InfProd_5133.png

InfProd_5134.png

InfProd_5135.png

InfProd_5136.png

InfProd_5137.png

InfProd_5138.png

InfProd_5139.png

InfProd_5140.png

InfProd_5141.png

InfProd_5142.png

InfProd_5143.png

InfProd_5144.png

InfProd_5145.png

InfProd_5146.png

InfProd_5147.png

InfProd_5148.png

InfProd_5149.png

InfProd_5150.png

InfProd_5151.png

InfProd_5152.png

InfProd_5153.png

InfProd_5154.png

InfProd_5155.png

InfProd_5156.png

InfProd_5157.png

InfProd_5158.png

InfProd_5159.png

InfProd_5160.png

InfProd_5161.png

InfProd_5162.png

InfProd_5163.png

InfProd_5164.png

InfProd_5165.png

InfProd_5166.png

InfProd_5167.gif

InfProd_5168.png

InfProd_5169.png

InfProd_5170.png

InfProd_5171.png

InfProd_5172.png

InfProd_5173.png

InfProd_5174.png

InfProd_5175.png

InfProd_5176.png

InfProd_5177.png

InfProd_5178.png

InfProd_5179.png

InfProd_5180.png

InfProd_5181.png

InfProd_5182.png

InfProd_5183.png

InfProd_5184.png

InfProd_5185.png

InfProd_5186.png

InfProd_5187.png

InfProd_5188.png

InfProd_5189.png

InfProd_5190.png

InfProd_5191.png

InfProd_5192.png

InfProd_5193.png

InfProd_5194.png

InfProd_5195.png

InfProd_5196.png

InfProd_5197.png

InfProd_5198.png

InfProd_5199.png

InfProd_5200.png

InfProd_5201.png

InfProd_5202.png

InfProd_5203.png

InfProd_5204.png

InfProd_5205.png

InfProd_5206.png

InfProd_5207.png

InfProd_5208.png

InfProd_5209.png

InfProd_5210.png

InfProd_5211.png

InfProd_5212.png

InfProd_5213.png

InfProd_5214.png

InfProd_5215.png

InfProd_5216.png

InfProd_5217.png

InfProd_5218.png

InfProd_5219.png

InfProd_5220.png

InfProd_5221.png

InfProd_5222.png

InfProd_5223.png

InfProd_5224.png

InfProd_5225.png

InfProd_5226.png

InfProd_5227.png

InfProd_5228.png

InfProd_5229.png

InfProd_5230.png

InfProd_5231.png

InfProd_5232.gif

InfProd_5233.png

InfProd_5234.png

InfProd_5235.png

InfProd_5236.png

InfProd_5237.png

InfProd_5238.png

InfProd_5239.png

InfProd_5240.png

InfProd_5241.png

InfProd_5242.png

InfProd_5243.png

InfProd_5244.png

InfProd_5245.png

InfProd_5246.png

InfProd_5247.png

InfProd_5248.png

InfProd_5249.png

InfProd_5250.png

InfProd_5251.png

InfProd_5252.png

InfProd_5253.png

InfProd_5254.png

InfProd_5255.png

InfProd_5256.png

InfProd_5257.png

InfProd_5258.png

InfProd_5259.png

InfProd_5260.png

InfProd_5261.png

InfProd_5262.png

InfProd_5263.png

InfProd_5264.png

InfProd_5265.png

InfProd_5266.png

InfProd_5267.png

InfProd_5268.png

InfProd_5269.png

InfProd_5270.png

InfProd_5271.png

InfProd_5272.png

InfProd_5273.png

InfProd_5274.png

InfProd_5275.png

InfProd_5276.png

InfProd_5277.png

InfProd_5278.png

InfProd_5279.png

InfProd_5280.png

InfProd_5281.png

InfProd_5282.png

InfProd_5283.png

InfProd_5284.png

InfProd_5285.png

InfProd_5286.png

InfProd_5287.png

InfProd_5288.png

InfProd_5289.png

InfProd_5290.png

InfProd_5291.png

InfProd_5292.png

InfProd_5293.png

InfProd_5294.png

InfProd_5295.png

InfProd_5296.png

InfProd_5297.png

InfProd_5298.png

InfProd_5299.png

InfProd_5300.png

InfProd_5301.png

InfProd_5302.png

InfProd_5303.png

InfProd_5304.png

InfProd_5305.png

InfProd_5306.png

InfProd_5307.png

InfProd_5308.png

InfProd_5309.png

InfProd_5310.png

InfProd_5311.png

InfProd_5312.png

InfProd_5313.png

InfProd_5314.png

InfProd_5315.png

InfProd_5316.png

InfProd_5317.png

InfProd_5318.png

InfProd_5319.png

InfProd_5320.png

InfProd_5321.png

InfProd_5322.png

InfProd_5323.png

InfProd_5324.png

InfProd_5325.png

InfProd_5326.png

InfProd_5327.png

InfProd_5328.png

InfProd_5329.png

InfProd_5330.png

InfProd_5331.png

InfProd_5332.png

InfProd_5333.png

InfProd_5334.png

InfProd_5335.png

InfProd_5336.png

InfProd_5337.png

InfProd_5338.png

InfProd_5339.png

InfProd_5340.png

InfProd_5341.png

InfProd_5342.png

InfProd_5343.png

InfProd_5344.png

InfProd_5345.png

InfProd_5346.png

InfProd_5347.png

InfProd_5348.png

InfProd_5349.png

InfProd_5350.png

InfProd_5351.png

InfProd_5352.png

InfProd_5353.png

InfProd_5354.png

InfProd_5355.png

InfProd_5356.png

InfProd_5357.png

InfProd_5358.png

InfProd_5359.png

InfProd_5360.png

InfProd_5361.png

InfProd_5362.png

InfProd_5363.png

InfProd_5364.png

InfProd_5365.png

InfProd_5366.png

InfProd_5367.png

InfProd_5368.png

InfProd_5369.png

InfProd_5370.png

InfProd_5371.png

InfProd_5372.png

InfProd_5373.png

InfProd_5374.png

InfProd_5375.png

InfProd_5376.png

InfProd_5377.png

InfProd_5378.png

InfProd_5379.png

InfProd_5380.png

InfProd_5381.png

InfProd_5382.png

InfProd_5383.png

InfProd_5384.png

InfProd_5385.png

InfProd_5386.png

InfProd_5387.png

InfProd_5388.png

InfProd_5389.png

InfProd_5390.png

InfProd_5391.png

InfProd_5392.png

InfProd_5393.png

InfProd_5394.png

InfProd_5395.png

InfProd_5396.png

InfProd_5397.png

InfProd_5398.png

InfProd_5399.png

InfProd_5400.png

InfProd_5401.png

InfProd_5402.png

InfProd_5403.png

InfProd_5404.png

InfProd_5405.png

InfProd_5406.png

InfProd_5407.png

InfProd_5408.png

InfProd_5409.png

InfProd_5410.png

InfProd_5411.png

InfProd_5412.png

InfProd_5413.png

InfProd_5414.png

InfProd_5415.png

InfProd_5416.png

InfProd_5417.png

InfProd_5418.png

InfProd_5419.png

InfProd_5420.png

InfProd_5421.png

InfProd_5422.png

InfProd_5423.png

InfProd_5424.png

InfProd_5425.png

InfProd_5426.png

InfProd_5427.png

InfProd_5428.png

InfProd_5429.png

InfProd_5430.png

InfProd_5431.png

InfProd_5432.png

InfProd_5433.png

InfProd_5434.png

InfProd_5435.png

InfProd_5436.png

InfProd_5437.png

InfProd_5438.png

InfProd_5439.png

InfProd_5440.png

InfProd_5441.png

InfProd_5442.png

InfProd_5443.png

InfProd_5444.png

InfProd_5445.png

InfProd_5446.png

InfProd_5447.gif

InfProd_5448.png

InfProd_5449.png

InfProd_5450.png

InfProd_5451.png

InfProd_5452.png

InfProd_5453.png

InfProd_5454.png

InfProd_5455.png

InfProd_5456.png

InfProd_5457.png

InfProd_5458.png

InfProd_5459.png

InfProd_5460.png

InfProd_5461.png

InfProd_5462.png

InfProd_5463.png

InfProd_5464.png

InfProd_5465.png

InfProd_5466.png

InfProd_5467.png

InfProd_5468.png

InfProd_5469.png

InfProd_5470.png

InfProd_5471.png

InfProd_5472.png

InfProd_5473.png

InfProd_5474.png

InfProd_5475.png

InfProd_5476.png

InfProd_5477.png

InfProd_5478.png

InfProd_5479.png

InfProd_5480.png

InfProd_5481.png

InfProd_5482.png

InfProd_5483.png

InfProd_5484.png

InfProd_5485.png

InfProd_5486.png

InfProd_5487.png

InfProd_5488.png

InfProd_5489.png

InfProd_5490.png

InfProd_5491.png

InfProd_5492.png

InfProd_5493.png

InfProd_5494.png

InfProd_5495.png

InfProd_5496.png

InfProd_5497.png

InfProd_5498.png

InfProd_5499.png

InfProd_5500.png

InfProd_5501.png

InfProd_5502.png

InfProd_5503.png

InfProd_5504.png

InfProd_5505.png

InfProd_5506.png

InfProd_5507.png

InfProd_5508.png

InfProd_5509.png

InfProd_5510.png

InfProd_5511.png

InfProd_5512.png

InfProd_5513.png

InfProd_5514.png

InfProd_5515.png

InfProd_5516.png

InfProd_5517.png

InfProd_5518.png

InfProd_5519.png

InfProd_5520.png

InfProd_5521.png

InfProd_5522.png

InfProd_5523.png

InfProd_5524.png

InfProd_5525.png

InfProd_5526.png

InfProd_5527.png

InfProd_5528.png

InfProd_5529.png

InfProd_5530.png

InfProd_5531.png

InfProd_5532.png

InfProd_5533.png

InfProd_5534.png

InfProd_5535.png

InfProd_5536.png

InfProd_5537.png

InfProd_5538.png

InfProd_5539.png

InfProd_5540.png

InfProd_5541.png

InfProd_5542.png

InfProd_5543.png

InfProd_5544.png

InfProd_5545.png

InfProd_5546.png

InfProd_5547.png

InfProd_5548.png

InfProd_5549.png

InfProd_5550.png

InfProd_5551.png

InfProd_5552.png

InfProd_5553.png

InfProd_5554.png

InfProd_5555.png

InfProd_5556.png

InfProd_5557.png

InfProd_5558.png

InfProd_5559.png

InfProd_5560.png

InfProd_5561.png

InfProd_5562.png

InfProd_5563.png

InfProd_5564.png

InfProd_5565.png

InfProd_5566.png

InfProd_5567.png

InfProd_5568.png

InfProd_5569.png

InfProd_5570.png

InfProd_5571.png

InfProd_5572.png

InfProd_5573.png

InfProd_5574.png

InfProd_5575.png

InfProd_5576.png

InfProd_5577.png

InfProd_5578.png

InfProd_5579.png

InfProd_5580.png

InfProd_5581.png

InfProd_5582.png

-(z→-z)

InfProd_5583.png

InfProd_5584.png

InfProd_5585.png

InfProd_5586.png

InfProd_5587.png

InfProd_5588.png

InfProd_5589.png

InfProd_5590.png

InfProd_5591.png

InfProd_5592.png

InfProd_5593.png

InfProd_5594.png

InfProd_5595.png

InfProd_5596.png

Proof double Product

InfProd_5597.png

InfProd_5598.png

InfProd_5599.png

InfProd_5600.png

InfProd_5601.png

InfProd_5602.png

InfProd_5603.png

InfProd_5604.png

InfProd_5605.png

InfProd_5606.png

InfProd_5607.png

InfProd_5608.png

InfProd_5609.png

InfProd_5610.png

InfProd_5611.png

InfProd_5612.png

InfProd_5613.png

InfProd_5614.png

InfProd_5615.png

InfProd_5616.png

InfProd_5617.png

InfProd_5618.png

InfProd_5619.png

InfProd_5620.png

InfProd_5621.png

InfProd_5622.png

InfProd_5623.png

InfProd_5624.png

InfProd_5625.png

InfProd_5626.png

InfProd_5627.png

InfProd_5628.png

InfProd_5629.png

InfProd_5630.png

InfProd_5631.png

InfProd_5632.png

InfProd_5633.png

InfProd_5634.png

InfProd_5635.png

InfProd_5636.png

InfProd_5637.png

InfProd_5638.png

InfProd_5639.png

InfProd_5640.png

InfProd_5641.png

InfProd_5642.png

InfProd_5643.png

InfProd_5644.png

InfProd_5645.png

InfProd_5646.gif

InfProd_5647.png

InfProd_5648.png

InfProd_5649.png

InfProd_5650.png

InfProd_5651.png

InfProd_5652.png

InfProd_5653.png

InfProd_5654.png

InfProd_5655.png

InfProd_5656.png

InfProd_5657.png

InfProd_5658.png

InfProd_5659.png

InfProd_5660.png

InfProd_5661.png

InfProd_5662.png

InfProd_5663.png

InfProd_5664.png

InfProd_5665.png

InfProd_5666.png

InfProd_5667.png

InfProd_5668.png

InfProd_5669.png

InfProd_5670.png

InfProd_5671.gif

Created with the Wolfram Language      Download Page    Indefinite Integrals     Definite Integrals