Collection of Infinite Products and Series

   Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

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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

-  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, Products_3.png, Products_4.png, Products_5.png, Products_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 )

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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 :

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Products of two Gammas :

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Partial Fraction Decompositions :

General expression :

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Special cases with n = 2 and m = 1 :

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  Products_60.png as simple function of k :

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Special cases with m = 0 :

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q - Product (0 < q < 1) :

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Two kinds of decomposition of the same product :

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With

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(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] :

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More Products :

The maximum of the next function is found at Products_77.png | f Products_78.png)) = (-1/2 | Products_79.png), its zeroes on the positive (negative) axis are the odd (even) Integers :

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Euler’s product :

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The next product approximates a Gauss function Products_109.png with Products_110.png :

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Products of trig functions :

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Products involving Theta Functions    ( Back to Top )

Products_161.png is shorthand for EllipticTheta[n, z, q] and Products_162.png means EllipticThetaPrime[m, z, q].

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

Series and Product Representations :

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With  Products_169.png[ 0 , q ] a few relations between the theta functions are

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

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Partial differential equation :

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EllipticThetas with imaginary argument :

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With z → 0 we get

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Half Lambda :

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Double Lambda :

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Other relations :

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Square and square root of q :

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

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Double product representation of the single theta functions :

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

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The theta functions may be expressed through each other :

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

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Trigonometric and hyperbolic Products :

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

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q - Products :

In the following is ( 0 < q < 1 ) and Products_251.png[ 0 , q ] ,   (Products_252.png[ 0 , q ] =Products_253.png[ 0 , - q ] ) :

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

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

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

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Products_293.pngProducts_294.png and Products_295.png can be expressed through m[q] , K[m[q]] and E[m[q]] :

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and similarly :

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and :

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and from combining the above like :

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we get :

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as q is getting larger than Products_301.png the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

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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 = Products_303.png) expressed through EllipticK and m :

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Products_325.png Products_326.png Products_327.png

Series expansion of InverseEllipticNomeQ :

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Series expansion of EllipticNomeQ :

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Specific Values :

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Products_351.png and Products_352.png are Ramanujans g functions, m = InverseEllipticNomeQ[Products_353.png]  (for each n the even g and the odd G seem to show a somewhat simpler structure than their counterparts) :

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products with q = Products_356.png :

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special cases :

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Theta Functions , specific values :

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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: Products_441.png means Products_442.png.

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Series involving exponentials :

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Series involving Products_520.png :

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Series of trig. functions :

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

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q - Series :   ( Back to Top )

with Products_651.png → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

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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.

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There is a small inconsistency in the definition of LerchPhi in the neighbourhood of a=0: Products_657.png.

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

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

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

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other :

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Products_903.png

Products_904.png

Products_905.png

Products_906.png

Products_907.png

Products_908.png

Products_909.png

QFunction Identities :

Products_910.png

Products_911.png

Products_912.png

Products_913.png

Products_914.png

Products_915.png

Products_916.png

Products_917.gif

Products_918.png

Products_919.png

Products_920.png

Products_921.png

Products_922.png

Products_923.png

Products_924.png

Products_925.png

Products_926.png

Products_927.png

Products_928.png

Products_929.png

Products_930.png

Products_931.png

Products_932.png

Products_933.png

Products_934.png

Products_935.png

Products_936.png

Products_937.png

Products_938.png

Products_939.png

Products_940.png

Products_941.png

Products_942.png

Products_943.png

Products_944.png

Products_945.png

Products_946.png

Products_947.png

Products_948.png

Products_949.png

Products_950.png

Products_951.png

Products_952.png

Products_953.png

Products_954.gif

Products_955.png

With x ∈ Reals is   Products_956.png

Products_957.png
Real Part Imaginary Part
n=1: Products_958.png 0
n=2: Products_959.png Products_960.png
n=3: Products_961.png Products_962.png
n=4: Products_963.png Products_964.png
n=5: Products_965.png Products_966.png
n=6: Products_967.png Products_968.png
n=7: Products_969.png Products_970.png

Products_971.png

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

Products_972.png

E[m] is EllipticE[m];

Products_973.png

Products_974.png

Products_975.png

Products_976.png

Products_977.png

Products_978.png

Products_979.png

Products_980.png

Products_981.png

Products_982.png

Series of Hyperbolic Functions:   ( Back to Top )

Products_983.png

Products_984.png

Products_985.png

Products_986.png

Products_987.png

Products_988.png

Products_989.png

Products_990.png

Products_991.png

Products_992.png

Products_993.png

Products_994.png

Products_995.png

Products_996.png

Products_997.png

Products_998.png

Products_999.png

Products_1000.png

Products_1001.png

Products_1002.png

Products_1003.png

Products_1004.png

Products_1005.png

Products_1006.png

Products_1007.png

Products_1008.png

Products_1009.png

ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: Products_1010.png means Products_1011.png.

Products_1012.png

Products_1013.png

Products_1014.png

Products_1015.png

Products_1016.png

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

Products_1017.png

Products_1018.png

Products_1019.png

Products_1020.png

Products_1021.png

Products_1022.png

m = InverseEllipticNomeQ[Products_1023.png] :

Products_1024.png

Products_1025.png

Products_1026.png

Products_1027.png

Products_1028.png

Products_1029.png

Products_1030.png

Products_1031.png

Products_1032.png

Products_1033.png

Products_1034.png

Products_1035.png

Products_1036.png

Products_1037.png

Products_1038.png

Products_1039.png

Products_1040.png

Products_1041.png

Products_1042.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” :

Products_1043.png

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

Products_1044.png

Products_1045.png

Products_1046.png

Products_1047.png

Products_1048.png

Products_1049.png

Products_1050.png

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

Products_1051.png

Products_1052.png

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

Products_1053.png

Products_1054.png

Products_1055.png

Products_1056.png

m = InverseEllipticNomeQ[Products_1057.png] :

Products_1058.png

Products_1059.png

Products_1060.png

Products_1061.png

Products_1062.png

Products_1063.png

Products_1064.png

Products_1065.png

Products_1066.png

Products_1067.png

Products_1068.png

Products_1069.png

Products_1070.png

Products_1071.png

Products_1072.png

Products_1073.png

Products_1074.png

Products_1075.png

Products_1076.png

Products_1077.png

Products_1078.png

Products_1079.png

Products_1080.png

Products_1081.png

Products_1082.png

Products_1083.png

Products_1084.png

Products_1085.png

Products_1086.png

Products_1087.png

Products_1088.png

Products_1089.png

m = InverseEllipticNomeQ[Products_1090.png] :

Products_1091.png

Products_1092.png

Products_1093.png

Products_1094.png

Products_1095.png

Some hyperbolic Identities :

Products_1096.png

Products_1097.png

Products_1098.gif

Products_1099.gif

Series of CosIntegral:   ( Back to Top )

Products_1100.png

Products_1101.png

Products_1102.png

Products_1103.png

Products_1104.png

Products_1105.png

Products_1106.png

Products_1107.png

Products_1108.png

Products_1109.png

Products_1110.png

Products_1111.png

Products_1112.png

Products_1113.png

Some Limits :   ( Back to Top )

Products_1114.png

Products_1115.png

Products_1116.png

Products_1117.png

Products_1118.png

Products_1119.png

Products_1120.png

diverse Series :   ( Back to Top )

Products_1121.png

Products_1122.png

Products_1123.png

Products_1124.png

Products_1125.png

Products_1126.png

Products_1127.png

Products_1128.png

Products_1129.png

Products_1130.png

Products_1131.png

Products_1132.png

Products_1133.png

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

Products_1134.png

Products_1135.png

Products_1136.png

Products_1137.png

Products_1138.png

Products_1139.png

Products_1140.png

Products_1141.png

Products_1142.png

Products_1143.png

Products_1144.png

Products_1145.png

Products_1146.png

Products_1147.png

Products_1148.png

Products_1149.png

Products_1150.png

Products_1151.png

Products_1152.png

Products_1153.png

Products_1154.png

Products_1155.png

Products_1156.png

Products_1157.png

Products_1158.png

Multiple Sums :

Products_1159.png

Products_1160.png

Products_1161.png

Products_1162.png

Products_1163.png

Products_1164.png

Products_1165.png

Products_1166.png

Products_1167.png

Borwein' s formula :

Products_1168.png

Benson' s formula:

Products_1169.gif

Products_1170.png

Products_1171.png

Products_1172.png

Products_1173.png

Products_1174.png

Products_1175.png

Products_1176.png

Products_1177.png

Products_1178.png

Products_1179.png

Products_1180.png

Products_1181.png

more series :

Products_1182.png

Products_1183.png

Products_1184.png

Products_1185.png

Products_1186.png

Sum of the inverse k-gonal numbers :

Products_1187.png

Value of the series for the first k :

Products_1188.png

The sum  Products_1189.png  gives following results for some rational s=p/q :

Products_1190.png

Products_1191.png

Products_1192.png

Products_1193.png

Products_1194.png

Products_1195.png

Products_1196.png

Products_1197.png

Products_1198.png

Products_1199.png

Products_1200.png

Products_1201.png

Products_1202.png

Products_1203.png

Products_1204.png

Products_1205.png

Products_1206.png

Products_1207.png

Products_1208.png

Products_1209.png

Products_1210.png

Products_1211.png

Products_1212.png

Products_1213.png

Products_1214.png

Products_1215.png

Products_1216.png

Products_1217.png

This sum alternates between ± π  for z ∈ N :

Products_1218.png

In the following 4 expressions b =Products_1219.png :

Products_1220.png

Products_1221.png

Products_1222.png

Products_1223.png

Products_1224.png

Products_1225.png

Products_1226.png

Products_1227.png

Products_1228.png

The next three expressions contain s = Products_1229.png and t = Products_1230.png:

Products_1231.png

Products_1232.png

Products_1233.png

Products_1234.png

Products_1235.png

Products_1236.png

Products_1237.png

Products_1238.png

Products_1239.png

Products_1240.png

Products_1241.png

Products_1242.png

Products_1243.png

Products_1244.png

Products_1245.png

Products_1246.png

Products_1247.png

Products_1248.png

Products_1249.png

Products_1250.png

Products_1251.png

Products_1252.png

Products_1253.png

Products_1254.png

Products_1255.png

Products_1256.png

Products_1257.png

Products_1258.png

Products_1259.png

Products_1260.png

Products_1261.png

Products_1262.png

Products_1263.png

Products_1264.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.

Products_1265.png

Products_1266.png

Products_1267.png

Products_1268.png

Products_1269.png

Products_1270.png

Products_1271.png

Products_1272.png

Products_1273.png

Products_1274.png

Products_1275.png

Products_1276.png

Products_1277.png

Products_1278.png

Products_1279.png

Products_1280.png

Products_1281.png

Products_1282.png

Products_1283.png

Products_1284.png

Products_1285.png

Products_1286.png

Products_1287.png

Products_1288.png

Products_1289.png

Products_1290.png

Products_1291.png

Products_1292.png

Products_1293.png

Products_1294.png

Products_1295.png

Products_1296.png

Products_1297.png

Products_1298.png

Products_1299.png

Products_1300.png

Products_1301.png

Products_1302.gif

Products_1303.png

Products_1304.png

Products_1305.png

Products_1306.png

Products_1307.png

Products_1308.png

Products_1309.png

Products_1310.png

Products_1311.png

Products_1312.png

Products_1313.png

Products_1314.png

Products_1315.png

Products_1316.png

Products_1317.png

Products_1318.png

Products_1319.png

Products_1320.png

Products_1321.png

Products_1322.png

Products_1323.png

Products_1324.png

Products_1325.png

Products_1326.png

Products_1327.png

Products_1328.png

Products_1329.png

Products_1330.png

Products_1331.png

Products_1332.png

Products_1333.png

Products_1334.png

Products_1335.png

Products_1336.png

Products_1337.png

Products_1338.png

Products_1339.png

Products_1340.png

Products_1341.png

Products_1342.png

Products_1343.png

Products_1344.png

Products_1345.png

Products_1346.png

Products_1347.png

Products_1348.png

Products_1349.png

Products_1350.png

Products_1351.png

Products_1352.png

Products_1353.png

Products_1354.png

Products_1355.png

Products_1356.png

Products_1357.png

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

Products_1358.png

Products_1359.png

Products_1360.png

Products_1361.gif

Products_1362.png

Products_1363.png

Products_1364.png

Products_1365.png

Products_1366.png

Products_1367.png

Products_1368.png

Products_1369.png

Products_1370.png

Products_1371.png

Products_1372.png

Products_1373.png

Products_1374.png

Products_1375.png

Products_1376.png

Products_1377.png

Products_1378.png

Products_1379.png

Products_1380.png

Products_1381.png

Products_1382.png

Products_1383.png

Products_1384.png

Products_1385.png

Products_1386.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.

Products_1387.png

Products_1388.png

Products_1389.png

Products_1390.png

Products_1391.png

Products_1392.png

Products_1393.png

Products_1394.png

Products_1395.png

Products_1396.png

Products_1397.png

Products_1398.png

Products_1399.png

Products_1400.png

Products_1401.png

Products_1402.png

Products_1403.png

Products_1404.png

Products_1405.gif

Products_1406.png

Products_1407.png

Products_1408.png

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

Products_1409.png

Products_1410.png

Products_1411.png

Products_1412.png

Products_1413.png

Products_1414.png

Products_1415.png

Products_1416.png

Products_1417.png

Products_1418.png

Products_1419.png

Products_1420.png

Products_1421.png

Products_1422.png

Products_1423.png

Products_1424.png

Products_1425.png

Products_1426.png

Products_1427.png

Products_1428.png

Products_1429.png

Products_1430.png

Products_1431.png

Products_1432.png

Products_1433.png

Products_1434.png

Products_1435.png

Products_1436.png

Products_1437.png

Products_1438.png

Products_1439.png

Products_1440.png

Products_1441.png

Products_1442.png

Products_1443.png

Products_1444.png

Products_1445.png

Products_1446.png

Some ArcTan Identities :

Products_1447.gif

Series of Bessel Functions :   ( Back to Top )

Products_1448.png

Products_1449.png

Products_1450.png

Products_1451.png

Products_1452.png

Products_1453.png

Products_1454.png

Products_1455.png

Products_1456.png

Products_1457.png

Products_1458.png

Products_1459.png

Products_1460.png

Products_1461.png

Products_1462.png

Products_1463.png

Products_1464.png

Products_1465.png

Products_1466.png

Products_1467.png

Products_1468.png

Products_1469.png

Products_1470.png

Products_1471.png

Products_1472.png

Products_1473.png

Products_1474.png

Products_1475.png

Products_1476.png

Products_1477.png

Products_1478.png

Products_1479.png

Products_1480.png

Products_1481.png

Products_1482.png

Products_1483.png

Products_1484.png

Products_1485.png

Products_1486.png

Products_1487.png

Products_1488.png

Products_1489.png

Products_1490.png

Products_1491.png

Products_1492.png

Products_1493.png

Products_1494.png

Products_1495.png

Products_1496.png

Products_1497.png

Products_1498.png

Products_1499.png

Products_1500.png

Products_1501.png

Products_1502.png

Products_1503.png

Products_1504.png

Products_1505.png

Products_1506.gif

Products_1507.png

Products_1508.png

Products_1509.png

Products_1510.png

Products_1511.png

Products_1512.png

Series of Legendre Polynomials :   ( Back to Top )

Products_1513.png

Products_1514.png

Products_1515.png

Products_1516.png

Products_1517.png

Products_1518.png

Products_1519.png

Products_1520.png

Products_1521.png

Products_1522.png

Series of Jacobi Polynomials :

Products_1523.png

Series of Hermite Polynomials :

Products_1524.png

Products_1525.png

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

Products_1526.png

Products_1527.png

Products_1528.png

Products_1529.png

Products_1530.png

Products_1531.png

Products_1532.png

Products_1533.png

Products_1534.png

Products_1535.png

Products_1536.png

Products_1537.png

Products_1538.png

Products_1539.png

Products_1540.png

Products_1541.png

Products_1542.png

Products_1543.png

Products_1544.png

Products_1545.png

Products_1546.png

Products_1547.png

Products_1548.png

Products_1549.png

Products_1550.png

Products_1551.png

Products_1552.png

Products_1553.png

Products_1554.png

Products_1555.png

Products_1556.png

Products_1557.png

Products_1558.png

Products_1559.png

Products_1560.gif

Special Values of Zeta :

Products_1561.png

Products_1562.png

PolyGamma :

Products_1563.png

Products_1564.png

Products_1565.png

Products_1566.png

Products_1567.png

Products_1568.png

Products_1569.png

Products_1570.png

Products_1571.png

Products_1572.png

Products_1573.png

Products_1574.png

Products_1575.png

Products_1576.png

Products_1577.png

Products_1578.png

Products_1579.png

Products_1580.png

Products_1581.png

Products_1582.png

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

Products_1583.png

PolyLog and  LerchPhi :

Products_1584.png

Products_1585.png

Products_1586.png

Products_1587.png

Products_1588.png

Products_1589.png

Products_1590.png

Products_1591.png

Products_1592.png

Products_1593.png

Products_1594.png

Products_1595.png

Products_1596.png

Products_1597.png

Products_1598.png

Products_1599.png

Products_1600.png

Products_1601.png

Products_1602.png

Products_1603.png

Products_1604.png

Products_1605.png

Products_1606.png

Products_1607.png

Products_1608.png

Products_1609.png

Products_1610.png

Products_1611.png

Products_1612.png

Products_1613.png

Products_1614.png

Products_1615.png

Products_1616.png

Products_1617.png

Products_1618.png

Products_1619.png

Products_1620.png

Products_1621.png

Products_1622.png

Products_1623.png

Products_1624.png

Products_1625.png

Products_1626.png

Special Values of PolyLog and LerchPhi :

Products_1627.png

Products_1628.png

Products_1629.png

Products_1630.png

Products_1631.png

Products_1632.png

Products_1633.png

Products_1634.png

Products_1635.png

Products_1636.png

Series of Beta Functions :   ( Back to Top )

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

Products_1637.png

Products_1638.png

Products_1639.png

Products_1640.png

Products_1641.png

Products_1642.png

Products_1643.png

Products_1644.png

Products_1645.png

Products_1646.png

Products_1647.png

Products_1648.png

Products_1649.png

Products_1650.png

Products_1651.png

Products_1652.png

Products_1653.png

Products_1654.png

Products_1655.png

Products_1656.png

Products_1657.png

Products_1658.png

Products_1659.png

Products_1660.png

Products_1661.png

Products_1662.png

Products_1663.png

Products_1664.png

Products_1665.png

Products_1666.png

Products_1667.png

Products_1668.png

Products_1669.png

Products_1670.png

Products_1671.png

Special values of InverseBetaRegularized :

Products_1672.png

Series of Gamma Functions :   ( Back to Top )

Products_1673.png

Dougall' s Formula :

Products_1674.png

Products_1675.png

Products_1676.png

Products_1677.gif

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

Products_1678.png

Products_1679.png

Products_1680.png

Products_1681.png

Products_1682.png

Products_1683.png

Products_1684.png

Products_1685.png

Products_1686.png

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

Products_1687.png

Products_1688.png

Products_1689.png

Products_1690.png

Products_1691.png

Products_1692.png

Products_1693.png

Special values of InverseGammaRegularized :

Products_1694.png

Series involving HarmonicNumber : ( Back To Top )

Products_1695.png

Products_1696.png

Products_1697.png

Products_1698.png

Products_1699.png

Products_1700.png

Products_1701.png

Products_1702.png

Products_1703.png

Products_1704.png

Products_1705.png

Products_1706.png

Products_1707.png

Products_1708.png

Products_1709.png

Products_1710.png

Products_1711.png

Products_1712.png

Products_1713.png

Products_1714.png

Products_1715.png

Products_1716.png

Products_1717.png

Products_1718.png

Products_1719.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.

Products_1720.png

Products_1721.png

Products_1722.png

Products_1723.png

Series involving Hypergeometric Functions : ( Back to Top )

Products_1724.png

Products_1725.png

Products_1726.png

Products_1727.png

Products_1728.png

Products_1729.png

Products_1730.png

Products_1731.png

Products_1732.png

Products_1733.png

Products_1734.png

Products_1735.png

Products_1736.png

Products_1737.png

Products_1738.png

Products_1739.png

Products_1740.png

Products_1741.png

Products_1742.png

Products_1743.png

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Products_1745.png

Products_1746.png

Products_1747.png

Products_1748.png

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Products_1754.png

Products_1755.png

Products_1756.png

Products_1757.png

Products_1758.png

Products_1759.png

Products_1760.png

Products_1761.png

Products_1762.png

Products_1763.png

Products_1764.png

A few Integrals :   ( Back to Top )

Products_1765.png

Substitute  Products_1766.png   and the Feynman - Hibbs Integral

Products_1767.png

Products_1768.png

Iterated Expressions  ( Tetration ) :   ( Back to Top )

Products_1769.png

Products_1770.png

Products_1771.png

Products_1772.png

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: Products_1773.png.
f[x] is not defined beyond the maximum of its inverse function Products_1774.png, namely  Products_1775.png< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < Products_1776.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 ] .

Products_1777.png

Products_1778.png

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

Products_1779.png

These carry over with a = 0 to PolyLog :

Products_1780.png

Products_1781.png

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

Products_1782.png

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

Products_1783.png

The complex decomposition of  Products_1784.png with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression :

Products_1785.png

explicitly for low s and b = 2 :

Products_1786.png Products_1787.png
Products_1788.png Products_1789.png
Products_1790.png Products_1791.png
Products_1792.png Products_1793.png
Products_1794.png Products_1795.png
Products_1796.png Products_1797.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) :

Products_1798.png

The real part of  Products_1799.pngwith 1 ≤ x ∈ R is also given by

Products_1800.png

For (b ∈ N) is

Products_1801.png

The real and imaginary parts of LerchPhi[ Products_1802.png, 2, 1/2 ] (on the unit circle) are

Products_1803.png

With Clausen type functions for LerchPhi defined as

Products_1804.png

Products_1805.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 Products_1806.pngProducts_1807.png (on the unit circle) are

Products_1808.png

the expressions for Products_1809.png with lowest s being

Products_1810.png Products_1811.png
Products_1812.png Products_1813.png
Products_1814.png Products_1815.png
Products_1816.png Products_1817.png
Products_1818.png Products_1819.png
Products_1820.png Products_1821.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

Products_1822.png

The function Products_1823.png has an interesting derivative :

Products_1824.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 :

Products_1825.png

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

Products_1826.png

With Clausen type functions defined as

Products_1827.png

Products_1828.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 Products_1829.png (on the unit circle) are

Products_1830.png

the expressions for Products_1831.png with lowest s being

Products_1832.png

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

Products_1833.png

As before the derivative Products_1834.pngProducts_1835.pngis Products_1836.png with lowered index.
The first non polynomial partnerfunctions are found to be

Products_1837.png

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

Products_1838.png

explicitly for low s :

Products_1839.png Products_1840.png
Products_1841.png Products_1842.png
Products_1843.png Products_1844.png
Products_1845.png Products_1846.png
Products_1847.png Products_1848.png
Products_1849.png Products_1850.png
Products_1851.png Products_1852.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 :

Products_1853.png

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

Products_1854.png

Products_1855.png

Products_1856.png

Products_1857.png

Products_1858.png

Products_1859.png

Products_1860.png

Products_1861.png

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
Products_1862.png Products_1863.png Products_1864.png
Products_1865.png Products_1866.png Products_1867.png
Products_1868.png Products_1869.png Products_1870.png
Products_1871.png Products_1872.png Products_1873.png
Products_1874.png Products_1875.png Products_1876.png
Products_1877.png Products_1878.png Products_1879.png

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

Products_1880.png

Connection to Bernoulli and Euler numbers :

Products_1881.png

Clausens Integral :

Products_1882.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