Collection of Infinite Products and Series

   Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

InfProd_1.png

InfProd_2.png

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

-  some Limits

-  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

-  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];
some of the products 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 following 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 (n) and are referenced below at the bottom.

Infinite Products : ( Back to Top )

InfProd_7.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 in z :

InfProd_8.png

InfProd_9.png

InfProd_10.png

InfProd_11.png

InfProd_12.png

InfProd_13.png

InfProd_14.png

InfProd_15.png

InfProd_16.png

InfProd_17.png

InfProd_18.png

InfProd_19.png

InfProd_20.png

InfProd_21.png

InfProd_22.png

InfProd_23.png

InfProd_24.png

InfProd_25.png

InfProd_26.png

InfProd_27.png

InfProd_28.png

InfProd_29.png

InfProd_30.png

InfProd_31.png

InfProd_32.gif

InfProd_33.png

InfProd_34.png

InfProd_35.png

InfProd_36.png

InfProd_37.png

InfProd_38.png

InfProd_39.png

InfProd_40.png

InfProd_41.png

InfProd_42.png

InfProd_43.png

InfProd_44.png

InfProd_45.png

InfProd_46.png

InfProd_47.png

InfProd_48.png

InfProd_49.png

InfProd_50.png

InfProd_51.png

InfProd_52.png

InfProd_53.png

InfProd_54.png

Products of two Gammas :

InfProd_55.png

InfProd_56.png

InfProd_57.png

Partial Fraction Decompositions :

General expression :

InfProd_58.png

Special cases with n = 2 and m = 1 :

InfProd_59.png

  InfProd_60.png as simple function of k :

InfProd_61.png

InfProd_62.png

InfProd_63.png

InfProd_64.png

InfProd_65.png

InfProd_66.png

Special cases with m = 0 :

InfProd_67.png

InfProd_68.png

InfProd_69.png

InfProd_70.png

InfProd_71.png

q - Product (0 < q < 1) :

InfProd_72.png

InfProd_73.png

Two kinds of decomposition of the same product :

InfProd_74.gif

With

InfProd_75.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_76.png

If f[k] = InfProd_77.png  and  0 < m then

InfProd_78.png

With any f  independent of k:

InfProd_79.png

More Products :

The maximum of the next function is found at InfProd_80.png | f InfProd_81.png)) = (-1/2 | InfProd_82.png), its zeroes on the positive (negative) axis are the odd (even) Integers :

InfProd_83.png

InfProd_84.png

InfProd_85.png

InfProd_86.png

InfProd_87.png

InfProd_88.png

InfProd_89.png

InfProd_90.png

InfProd_91.png

InfProd_92.png

InfProd_93.png

InfProd_94.png

InfProd_95.png

InfProd_96.png

InfProd_97.png

InfProd_98.png

InfProd_99.png

InfProd_100.png

InfProd_101.png

InfProd_102.png

Euler’s product :

InfProd_103.png

InfProd_104.png

InfProd_105.png

InfProd_106.png

InfProd_107.png

InfProd_108.png

InfProd_109.png

InfProd_110.png

InfProd_111.png

The next product approximates a Gauss function InfProd_112.png with InfProd_113.png :

InfProd_114.gif

InfProd_115.png

InfProd_116.png

InfProd_117.png

InfProd_118.png

InfProd_119.png

InfProd_120.png

InfProd_121.png

InfProd_122.png

Products of trig functions :

InfProd_123.png

InfProd_124.png

InfProd_125.png

InfProd_126.png

InfProd_127.png

InfProd_128.png

InfProd_129.png

InfProd_130.png

InfProd_131.png

InfProd_132.png

InfProd_133.png

InfProd_134.png

InfProd_135.png

InfProd_136.png

InfProd_137.png

InfProd_138.png

InfProd_139.png

InfProd_140.png

InfProd_141.png

InfProd_142.png

InfProd_143.png

InfProd_144.png

InfProd_145.png

InfProd_146.png

InfProd_147.png

InfProd_148.png

InfProd_149.gif

InfProd_150.png

InfProd_151.png

InfProd_152.png

InfProd_153.png

InfProd_154.png

InfProd_155.png

InfProd_156.png

InfProd_157.png

InfProd_158.png

InfProd_159.png

InfProd_160.png

InfProd_161.png

InfProd_162.png

InfProd_163.png

Products involving Theta Functions    ( Back to Top )

InfProd_164.png is shorthand for EllipticTheta[n, z, q] and InfProd_165.png means EllipticThetaPrime[m, z, q].

InfProd_166.png

( The above product numerically converges best if k ≫ n. )

Series and Product Representations :

InfProd_167.png

InfProd_168.png

InfProd_169.png

InfProd_170.png

InfProd_171.png

With  InfProd_172.png[ 0 , q ] a few relations between the theta functions are

InfProd_173.gif

InfProd_174.png

InfProd_175.png

InfProd_176.png

InfProd_177.png

InfProd_178.png

InfProd_179.png

InfProd_180.png

InfProd_181.png

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

InfProd_182.png

Partial differential equation :

InfProd_183.png

EllipticThetas with imaginary argument :

InfProd_184.png

With z → 0 we get

InfProd_185.png

Half Lambda :

InfProd_186.png

InfProd_187.png

InfProd_188.png

InfProd_189.png

InfProd_190.png

InfProd_191.png

Double Lambda :

InfProd_192.png

InfProd_193.png

Other relations :

InfProd_194.png

Square and square root of q :

InfProd_195.png

InfProd_196.png

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

InfProd_197.png

InfProd_198.png

Double product representation of the single theta functions :

InfProd_199.png

InfProd_200.png

InfProd_201.png

InfProd_202.png

If the product over k is carried out first we get products with Tanh and Coth :

InfProd_203.png

InfProd_204.png

InfProd_205.png

InfProd_206.png

InfProd_207.png

InfProd_208.png

InfProd_209.png

InfProd_210.png

The theta functions may be expressed through each other :

InfProd_211.png

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

InfProd_212.png

Trigonometric and hyperbolic Products :

InfProd_213.png

InfProd_214.png

InfProd_215.png

InfProd_216.png

InfProd_217.png

InfProd_218.png

InfProd_219.png

InfProd_220.png

InfProd_221.png

InfProd_222.png

InfProd_223.png

InfProd_224.png

InfProd_225.png

InfProd_226.png

InfProd_227.png

InfProd_228.png

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

InfProd_229.png

InfProd_230.png

InfProd_231.png

InfProd_232.png

InfProd_233.png

InfProd_234.png

InfProd_235.png

InfProd_236.png

InfProd_237.png

InfProd_238.png

InfProd_239.png

InfProd_240.png

InfProd_241.png

InfProd_242.png

InfProd_243.png

InfProd_244.png

InfProd_245.png

InfProd_246.png

InfProd_247.png

InfProd_248.png

InfProd_249.png

InfProd_250.png

InfProd_251.png

InfProd_252.png

InfProd_253.png

q - Products :

In the following is ( 0 < q < 1 ) and InfProd_254.png[ 0 , q ] ,   (InfProd_255.png[ 0 , q ] =InfProd_256.png[ 0 , - q ] ) :

InfProd_257.png

InfProd_258.png

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

InfProd_259.png

InfProd_260.gif

InfProd_261.png

InfProd_262.png

InfProd_263.png

InfProd_264.png

InfProd_265.png

InfProd_266.png

InfProd_267.png

InfProd_268.png

InfProd_269.png

InfProd_270.png

InfProd_271.png

InfProd_272.png

InfProd_273.png

InfProd_274.png

InfProd_275.png

InfProd_276.png

InfProd_277.png

InfProd_278.png

InfProd_279.png

InfProd_280.png

InfProd_281.png

InfProd_282.png

InfProd_283.png

InfProd_284.png

InfProd_285.png

InfProd_286.png

InfProd_287.png

InfProd_288.png

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

InfProd_289.png

InfProd_290.png

InfProd_291.png

InfProd_292.png

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

InfProd_293.gif

InfProd_294.png

InfProd_295.png

InfProd_296.png

InfProd_297.pngInfProd_298.png and InfProd_299.png can be expressed through m[q] , K[m[q]] and E[m[q]] :

InfProd_300.png

and similarly :

InfProd_301.png

and :

InfProd_302.png

and from combining the above like :

InfProd_303.png

we get :

InfProd_304.png

as q is getting larger than InfProd_305.png the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

InfProd_306.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 i 2 π

Theta Functions (z = InfProd_307.png) expressed through EllipticK and m :

InfProd_308.png InfProd_309.png InfProd_310.png
InfProd_311.png InfProd_312.png InfProd_313.png
InfProd_314.png InfProd_315.png InfProd_316.png
InfProd_317.png InfProd_318.png InfProd_319.png
InfProd_320.png InfProd_321.png InfProd_322.png
InfProd_323.png InfProd_324.png InfProd_325.png
InfProd_326.png InfProd_327.png InfProd_328.png
InfProd_329.png InfProd_330.png InfProd_331.png

Series expansion of InverseEllipticNomeQ :

InfProd_332.png

InfProd_333.png

InfProd_334.png

Series expansion of EllipticNomeQ :

InfProd_335.png

InfProd_336.png

Specific Values :

InfProd_337.png

InfProd_338.png

InfProd_339.png

InfProd_340.png

InfProd_341.png

InfProd_342.png

InfProd_343.png

InfProd_344.png

InfProd_345.png

InfProd_346.png

InfProd_347.png

InfProd_348.png

InfProd_349.png

InfProd_350.png

InfProd_351.png

InfProd_352.png

InfProd_353.png

InfProd_354.png

InfProd_355.png

InfProd_356.png and InfProd_357.png are Ramanujans g functions, m = InverseEllipticNomeQ[InfProd_358.png]  (for each n the even g and the odd G seem to show a somewhat simpler structure than their counterparts) :

InfProd_359.png

InfProd_360.gif

products with q = InfProd_361.png :

InfProd_362.png

InfProd_363.png

special cases :

InfProd_364.png

InfProd_365.png

InfProd_366.png

InfProd_367.png

InfProd_368.png

InfProd_369.png

InfProd_370.png

InfProd_371.png

InfProd_372.png

InfProd_373.png

InfProd_374.png

InfProd_375.png

InfProd_376.png

InfProd_377.png

InfProd_378.png

InfProd_379.png

InfProd_380.png

InfProd_381.png

InfProd_382.png

InfProd_383.png

InfProd_384.png

InfProd_385.png

InfProd_386.png

InfProd_387.png

InfProd_388.png

InfProd_389.png

InfProd_390.png

InfProd_391.png

Theta Functions , specific values :

InfProd_392.png

InfProd_393.png

InfProd_394.png

InfProd_395.gif

InfProd_396.gif

InfProd_397.png

InfProd_398.gif

InfProd_399.gif

InfProd_400.png

InfProd_401.png

InfProd_402.png

InfProd_403.png

InfProd_404.png

InfProd_405.png

InfProd_406.png

InfProd_407.png

InfProd_408.png

InfProd_409.png

InfProd_410.png

InfProd_411.png

InfProd_412.png

InfProd_413.png

InfProd_414.png

InfProd_415.png

InfProd_416.png

InfProd_417.png

InfProd_418.png

InfProd_419.png

InfProd_420.png

InfProd_421.png

InfProd_422.gif

InfProd_423.png

InfProd_424.png

InfProd_425.png

InfProd_426.png

InfProd_427.png

InfProd_428.png

InfProd_429.png

InfProd_430.png

InfProd_431.png

InfProd_432.png

InfProd_433.png

InfProd_434.png

InfProd_435.png

InfProd_436.png

InfProd_437.png

InfProd_438.png

InfProd_439.png

InfProd_440.png

InfProd_441.png

InfProd_442.png

InfProd_443.png

InfProd_444.png

InfProd_445.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_446.png means InfProd_447.png.

InfProd_448.png

InfProd_449.png

InfProd_450.png

InfProd_451.png

InfProd_452.png

InfProd_453.png

InfProd_454.png

InfProd_455.png

InfProd_456.png

InfProd_457.png

InfProd_458.png

InfProd_459.png

InfProd_460.png

InfProd_461.png

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

Series involving exponentials :

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

InfProd_494.png

InfProd_495.png

InfProd_496.png

InfProd_497.png

InfProd_498.png

InfProd_499.png

InfProd_500.png

InfProd_501.png

InfProd_502.png

InfProd_503.png

InfProd_504.png

InfProd_505.png

InfProd_506.png

InfProd_507.png

InfProd_508.png

InfProd_509.png

InfProd_510.png

InfProd_511.png

InfProd_512.png

InfProd_513.png

InfProd_514.png

InfProd_515.png

InfProd_516.png

InfProd_517.png

InfProd_518.png

InfProd_519.png

InfProd_520.png

InfProd_521.png

InfProd_522.png

InfProd_523.png

InfProd_524.png

Series involving InfProd_525.png :

InfProd_526.gif

InfProd_527.png

InfProd_528.png

InfProd_529.png

InfProd_530.png

InfProd_531.png

InfProd_532.png

InfProd_533.png

InfProd_534.png

InfProd_535.png

InfProd_536.png

InfProd_537.png

InfProd_538.png

InfProd_539.png

InfProd_540.png

InfProd_541.png

InfProd_542.png

InfProd_543.png

Series of trig. functions :

InfProd_544.png

InfProd_545.png

InfProd_546.png

InfProd_547.png

InfProd_548.png

InfProd_549.png

InfProd_550.png

InfProd_551.png

InfProd_552.png

InfProd_553.png

InfProd_554.png

InfProd_555.png

InfProd_556.png

InfProd_557.png

InfProd_558.png

InfProd_559.png

InfProd_560.png

InfProd_561.png

InfProd_562.png

InfProd_563.png

InfProd_564.png

InfProd_565.png

InfProd_566.png

InfProd_567.png

InfProd_568.png

InfProd_569.png

InfProd_570.png

InfProd_571.png

InfProd_572.png

InfProd_573.png

InfProd_574.png

InfProd_575.png

InfProd_576.png

InfProd_577.png

InfProd_578.png

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

InfProd_588.png

InfProd_589.png

InfProd_590.png

InfProd_591.png

InfProd_592.png

InfProd_593.png

InfProd_594.png

InfProd_595.png

InfProd_596.png

InfProd_597.png

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

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

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.gif

InfProd_620.png

InfProd_621.png

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

InfProd_652.png

InfProd_653.png

InfProd_654.png

InfProd_655.png

InfProd_656.png

InfProd_657.png

q - Series :   ( Back to Top )

with InfProd_658.png → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

InfProd_659.png

The inner sum above gives the number of ascending sequences of length k in the permutations of n numbers.
For natural n PolyLog[-n, q] appears as a rational function in q.

InfProd_660.png

InfProd_661.png

InfProd_662.png

InfProd_663.png

There is a small inconsistency in the definition of LerchPhi in the neighbourhood of a=0: InfProd_664.png.

InfProd_665.png

InfProd_666.png

InfProd_667.png

InfProd_668.png

InfProd_669.png

InfProd_670.png

InfProd_671.png

InfProd_672.png

InfProd_673.png

InfProd_674.png

InfProd_675.gif

InfProd_676.gif

InfProd_677.png

InfProd_678.gif

InfProd_679.png

InfProd_680.png

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

InfProd_681.png

InfProd_682.png

InfProd_683.png

InfProd_684.png

InfProd_685.png

InfProd_686.png

InfProd_687.png

InfProd_688.png

InfProd_689.png

InfProd_690.png

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

InfProd_691.png

InfProd_692.png

InfProd_693.png

InfProd_694.gif

InfProd_695.png

InfProd_696.gif

InfProd_697.gif

InfProd_698.png

InfProd_699.png

InfProd_700.gif

InfProd_701.gif

InfProd_702.gif

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

InfProd_703.png

InfProd_704.png

InfProd_705.png

InfProd_706.png

InfProd_707.png

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

InfProd_737.png

InfProd_738.png

InfProd_739.png

InfProd_740.png

InfProd_741.png

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

InfProd_772.png

InfProd_773.png

InfProd_774.png

InfProd_775.png

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.png

InfProd_791.png

InfProd_792.png

InfProd_793.png

InfProd_794.png

InfProd_795.png

InfProd_796.png

InfProd_797.png

InfProd_798.png

InfProd_799.png

InfProd_800.png

InfProd_801.png

InfProd_802.png

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.png

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

other :

InfProd_898.png

InfProd_899.png

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.png

InfProd_917.png

InfProd_918.png

QFunction Identities :

InfProd_919.png

InfProd_920.png

InfProd_921.png

InfProd_922.png

InfProd_923.png

InfProd_924.png

InfProd_925.png

InfProd_926.gif

InfProd_927.png

InfProd_928.png

InfProd_929.png

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.png

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.png

InfProd_954.png

InfProd_955.png

InfProd_956.png

InfProd_957.png

InfProd_958.png

InfProd_959.png

InfProd_960.png

InfProd_961.png

InfProd_962.png

InfProd_963.gif

InfProd_964.png

With x ∈ Reals is   InfProd_965.png

InfProd_966.png
Real Part Imaginary Part
n=1: InfProd_967.png 0
n=2: InfProd_968.png InfProd_969.png
n=3: InfProd_970.png InfProd_971.png
n=4: InfProd_972.png InfProd_973.png
n=5: InfProd_974.png InfProd_975.png
n=6: InfProd_976.png InfProd_977.png
n=7: InfProd_978.png InfProd_979.png

InfProd_980.png

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

InfProd_981.png

E[m] is EllipticE[m];

InfProd_982.png

InfProd_983.png

InfProd_984.png

InfProd_985.png

InfProd_986.png

InfProd_987.png

InfProd_988.png

InfProd_989.png

InfProd_990.png

InfProd_991.png

Series of Hyperbolic Functions:   ( Back to Top )

InfProd_992.png

InfProd_993.png

InfProd_994.png

InfProd_995.png

InfProd_996.png

InfProd_997.png

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

InfProd_1009.png

InfProd_1010.png

InfProd_1011.png

InfProd_1012.png

InfProd_1013.png

InfProd_1014.png

InfProd_1015.png

InfProd_1016.png

InfProd_1017.png

InfProd_1018.png

ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_1019.png means InfProd_1020.png.

InfProd_1021.png

InfProd_1022.png

InfProd_1023.png

InfProd_1024.png

InfProd_1025.png

For numerical checks the finite lower negative limit should be increased by 1 where the running index is shifted by - 1/2.

InfProd_1026.png

InfProd_1027.png

InfProd_1028.png

InfProd_1029.png

InfProd_1030.png

InfProd_1031.png

m = InverseEllipticNomeQ[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

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

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_1054.png

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

InfProd_1055.png

InfProd_1056.png

InfProd_1057.png

InfProd_1058.png

InfProd_1059.png

InfProd_1060.png

InfProd_1061.png

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

InfProd_1062.png

InfProd_1063.png

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

InfProd_1064.png

InfProd_1065.png

InfProd_1066.png

InfProd_1067.png

m = InverseEllipticNomeQ[InfProd_1068.png] :

InfProd_1069.png

InfProd_1070.png

InfProd_1071.png

InfProd_1072.png

InfProd_1073.png

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

InfProd_1099.png

InfProd_1100.png

m = InverseEllipticNomeQ[InfProd_1101.png] :

InfProd_1102.png

InfProd_1103.png

InfProd_1104.png

InfProd_1105.png

InfProd_1106.png

Some hyperbolic Identities :

InfProd_1107.png

InfProd_1108.png

InfProd_1109.gif

InfProd_1110.gif

Series of CosIntegral:   ( Back to Top )

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

InfProd_1121.png

InfProd_1122.png

InfProd_1123.png

InfProd_1124.png

Some Limits :   ( Back to Top )

InfProd_1125.png

InfProd_1126.png

InfProd_1127.png

InfProd_1128.png

InfProd_1129.png

InfProd_1130.png

InfProd_1131.png

diverse Series :   ( Back to Top )

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

InfProd_1142.png

InfProd_1143.png

InfProd_1144.png

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

InfProd_1145.png

InfProd_1146.png

InfProd_1147.png

InfProd_1148.png

InfProd_1149.png

InfProd_1150.png

InfProd_1151.png

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

InfProd_1162.png

InfProd_1163.png

InfProd_1164.png

InfProd_1165.png

InfProd_1166.png

InfProd_1167.png

InfProd_1168.png

InfProd_1169.png

Multiple Sums :

InfProd_1170.png

InfProd_1171.png

InfProd_1172.png

InfProd_1173.png

InfProd_1174.png

InfProd_1175.png

InfProd_1176.png

InfProd_1177.png

InfProd_1178.png

Borwein' s formula :

InfProd_1179.png

Benson' s formula:

InfProd_1180.gif

InfProd_1181.png

InfProd_1182.png

InfProd_1183.png

InfProd_1184.png

InfProd_1185.png

InfProd_1186.png

InfProd_1187.png

InfProd_1188.png

InfProd_1189.png

InfProd_1190.png

InfProd_1191.png

InfProd_1192.png

more series :

InfProd_1193.png

InfProd_1194.png

InfProd_1195.png

InfProd_1196.png

InfProd_1197.png

Sum of the inverse k-gonal numbers :

InfProd_1198.png

Value of the series for the first k :

InfProd_1199.png

The sum  InfProd_1200.png  gives following results for some rational s=p/q :

InfProd_1201.png

InfProd_1202.png

InfProd_1203.png

InfProd_1204.png

InfProd_1205.png

InfProd_1206.png

InfProd_1207.png

InfProd_1208.png

InfProd_1209.png

InfProd_1210.png

InfProd_1211.png

InfProd_1212.png

InfProd_1213.png

InfProd_1214.png

InfProd_1215.png

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

InfProd_1225.png

InfProd_1226.png

InfProd_1227.png

InfProd_1228.png

This sum alternates between ± π  for z ∈ N :

InfProd_1229.png

In the following 4 expressions b =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

The next three expressions contain s = InfProd_1240.png and t = InfProd_1241.png:

InfProd_1242.png

InfProd_1243.png

InfProd_1244.png

InfProd_1245.png

InfProd_1246.png

InfProd_1247.png

InfProd_1248.png

InfProd_1249.png

InfProd_1250.png

InfProd_1251.png

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.png

InfProd_1264.png

InfProd_1265.png

InfProd_1266.png

InfProd_1267.png

InfProd_1268.png

InfProd_1269.png

InfProd_1270.png

InfProd_1271.png

InfProd_1272.png

InfProd_1273.png

InfProd_1274.png

InfProd_1275.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 ∈ Integer.

InfProd_1276.png

InfProd_1277.png

InfProd_1278.png

InfProd_1279.png

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

InfProd_1291.png

InfProd_1292.png

InfProd_1293.png

InfProd_1294.png

InfProd_1295.png

InfProd_1296.png

InfProd_1297.png

InfProd_1298.png

InfProd_1299.png

InfProd_1300.png

InfProd_1301.png

InfProd_1302.png

InfProd_1303.png

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.gif

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

InfProd_1342.png

InfProd_1343.png

InfProd_1344.png

InfProd_1345.png

InfProd_1346.png

InfProd_1347.png

InfProd_1348.png

InfProd_1349.png

InfProd_1350.png

InfProd_1351.png

InfProd_1352.png

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

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

InfProd_1369.png

InfProd_1370.png

InfProd_1371.png

InfProd_1372.gif

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

InfProd_1396.png

InfProd_1397.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 ∈ Integer.

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.gif

InfProd_1417.png

InfProd_1418.png

InfProd_1419.png

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

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

Some ArcTan Identities :

InfProd_1458.gif

Series of Bessel Functions :   ( Back to Top )

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.gif

InfProd_1518.png

InfProd_1519.png

InfProd_1520.png

InfProd_1521.png

InfProd_1522.png

InfProd_1523.png

Series of Legendre Polynomials :   ( Back to Top )

InfProd_1524.png

InfProd_1525.png

InfProd_1526.png

InfProd_1527.png

InfProd_1528.png

InfProd_1529.png

InfProd_1530.png

InfProd_1531.png

InfProd_1532.png

InfProd_1533.png

InfProd_1534.png

Series of Jacobi Polynomials :

InfProd_1535.png

Series of Hermite Polynomials :

InfProd_1536.png

InfProd_1537.png

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

InfProd_1538.png

InfProd_1539.png

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

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.gif

Special Values of Zeta :

InfProd_1573.png

InfProd_1574.png

PolyGamma :

InfProd_1575.png

InfProd_1576.png

InfProd_1577.png

InfProd_1578.png

InfProd_1579.png

InfProd_1580.png

InfProd_1581.png

InfProd_1582.png

InfProd_1583.png

InfProd_1584.png

InfProd_1585.png

InfProd_1586.png

InfProd_1587.png

InfProd_1588.png

InfProd_1589.png

InfProd_1590.png

InfProd_1591.png

InfProd_1592.png

InfProd_1593.png

InfProd_1594.png

Take Gauss Multiplication formula of Gamma functions, apply D[Log[#], z] on both sides (#)  m + 1 times gives

InfProd_1595.png

PolyLog and  LerchPhi :

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.png

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

Special Values of PolyLog and LerchPhi :

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

Series of Beta Functions :   ( Back to Top )

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

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.png

InfProd_1662.png

InfProd_1663.png

InfProd_1664.png

InfProd_1665.png

InfProd_1666.png

InfProd_1667.png

InfProd_1668.png

InfProd_1669.png

InfProd_1670.png

InfProd_1671.png

InfProd_1672.png

InfProd_1673.png

InfProd_1674.png

InfProd_1675.png

InfProd_1676.png

InfProd_1677.png

InfProd_1678.png

InfProd_1679.png

InfProd_1680.png

InfProd_1681.png

InfProd_1682.png

Special values of InverseBetaRegularized :

InfProd_1683.png

Series of Gamma Functions :   ( Back to Top )

InfProd_1684.png

Dougall' s Formula :

InfProd_1685.png

InfProd_1686.png

InfProd_1687.png

InfProd_1688.gif

( K[x] = EllipticK[x], E[x] = EllipticE[x] ) :

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

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

InfProd_1698.png

InfProd_1699.png

InfProd_1700.png

InfProd_1701.png

InfProd_1702.png

InfProd_1703.png

InfProd_1704.png

Special values of InverseGammaRegularized :

InfProd_1705.png

Series involving HarmonicNumber : ( Back To Top )

InfProd_1706.png

InfProd_1707.png

InfProd_1708.png

InfProd_1709.png

InfProd_1710.png

InfProd_1711.png

InfProd_1712.png

InfProd_1713.png

InfProd_1714.png

InfProd_1715.png

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

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.

InfProd_1732.png

InfProd_1733.png

InfProd_1734.png

InfProd_1735.png

Series involving Hypergeometric Functions : ( Back to Top )

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

InfProd_1748.png

InfProd_1749.png

InfProd_1750.png

InfProd_1751.png

InfProd_1752.png

InfProd_1753.png

InfProd_1754.png

InfProd_1755.png

InfProd_1756.png

InfProd_1757.png

InfProd_1758.png

InfProd_1759.png

InfProd_1760.png

InfProd_1761.png

InfProd_1762.png

InfProd_1763.png

InfProd_1764.png

InfProd_1765.png

InfProd_1766.png

InfProd_1767.png

InfProd_1768.png

InfProd_1769.png

InfProd_1770.png

InfProd_1771.png

InfProd_1772.png

InfProd_1773.png

InfProd_1774.png

InfProd_1775.png

InfProd_1776.png

A few Integrals :   ( Back to Top )

InfProd_1777.png

Substitute  InfProd_1778.png   and the Feynman - Hibbs Integral

InfProd_1779.png

InfProd_1780.png

Iterated Expressions  ( Tetration ) :   ( Back to Top )

InfProd_1781.png

InfProd_1782.png

InfProd_1783.png

InfProd_1784.png

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: InfProd_1785.png.
f[x] is not defined beyond the maximum of its inverse function InfProd_1786.png, namely  InfProd_1787.png< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < InfProd_1788.png as well, where it shows a bifurcation.

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_1789.png

InfProd_1790.png

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

InfProd_1791.png

These carry over with a = 0 to PolyLog :

InfProd_1792.png

InfProd_1793.png

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

InfProd_1794.png

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

InfProd_1795.png

The complex decomposition of  InfProd_1796.png with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression :

InfProd_1797.png

explicitly for low s and b = 2 :

InfProd_1798.png InfProd_1799.png
InfProd_1800.png 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

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 makes a '+' in case of an imaginary part of z larger than zero, a '-' in all other cases) :

InfProd_1810.png

The real part of  InfProd_1811.pngwith 1 ≤ x ∈ R is also given by

InfProd_1812.png

For (b ∈ N) is

InfProd_1813.png

The real and imaginary parts of LerchPhi[ InfProd_1814.png, 2, 1/2 ] (on the unit circle) are

InfProd_1815.png

With Clausen type functions for LerchPhi defined as

InfProd_1816.png

InfProd_1817.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_1818.pngInfProd_1819.png (on the unit circle) are

InfProd_1820.png

the expressions for InfProd_1821.png with lowest s being

InfProd_1822.png InfProd_1823.png
InfProd_1824.png InfProd_1825.png
InfProd_1826.png InfProd_1827.png
InfProd_1828.png InfProd_1829.png
InfProd_1830.png InfProd_1831.png
InfProd_1832.png InfProd_1833.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_1834.png

The function InfProd_1835.png has an interesting derivative :

InfProd_1836.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_1837.png

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

InfProd_1838.png

With Clausen type functions defined as

InfProd_1839.png

InfProd_1840.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_1841.png (on the unit circle) are

InfProd_1842.png

the expressions for InfProd_1843.png with lowest s being

InfProd_1844.png

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

InfProd_1845.png

As before the derivative InfProd_1846.pngInfProd_1847.pngis InfProd_1848.png with lowered index.
The first non polynomial partnerfunctions are found to be

InfProd_1849.png

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

InfProd_1850.png

explicitly for low s :

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

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

InfProd_1865.png

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

InfProd_1866.png

InfProd_1867.png

InfProd_1868.png

InfProd_1869.png

InfProd_1870.png

InfProd_1871.png

InfProd_1872.png

InfProd_1873.png

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
InfProd_1874.png InfProd_1875.png InfProd_1876.png
InfProd_1877.png InfProd_1878.png InfProd_1879.png
InfProd_1880.png InfProd_1881.png InfProd_1882.png
InfProd_1883.png InfProd_1884.png InfProd_1885.png
InfProd_1886.png InfProd_1887.png InfProd_1888.png
InfProd_1889.png InfProd_1890.png InfProd_1891.png

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

InfProd_1892.png

Connection to Bernoulli and Euler numbers :

InfProd_1893.png

Clausens Integral :

InfProd_1894.png

Contributors :

(1) Udo Ausserlechner, Infineon, per email

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

(3) Stephen, per email

Created with the Wolfram Language      Download Page    Indefinite Integrals     Definite Integrals