KleinInvariant Notations Traditional name Klein invariant modular function Traditional notation z Mathematica StandardForm notation KleinInvariant z Primary definition 09.50.0.0001.01 ϑ 0, Π z 8 ϑ 3 0, Π z 8 ϑ 4 0, Π z 8 3 ; Im z 0 54 ϑ 0, Π z ϑ 3 0, Π z ϑ 4 0, Π z 8 This definition can be continued to selected points on the real axis. Specific values Specialized values 09.50.03.0001.01 m 1 ; m Values at fixed points Values at complex z 09.50.03.000.01 1 Values at quadratic irrationalities (Heegner Numbers) 09.50.03.0013.01 1 3 1
http://functions.wolfram.com 1 3 1 7 1 11 1 19 1 3 3 1 43 1 67 1 163 09.50.03.0004.01 0 09.50.03.0005.01 15 64 09.50.03.0006.01 51 7 09.50.03.0007.01 51 09.50.03.0008.01 64 000 9 09.50.03.0009.01 51 000 09.50.03.0010.01 85 184 000 09.50.03.0011.01 151 931 373 056 000 Values at infinities 09.50.03.001.01 General characteristics Domain and analyticity z is an analytical function of z which is defined over the upper half of the complex z-plane. 09.50.04.0001.01 z z Symmetries and periodicities Periodicity
http://functions.wolfram.com 3 09.50.04.000.01 m z ; m Poles and essential singularities On the boundary of analyticity the function z has a dense set of poles. 09.50.04.0003.01 ing z z ; Im z 0 Branch points The function z does not have branch points. 09.50.04.0004.01 z z Branch cuts The function z does not have branch cuts. 09.50.04.0005.01 z z Natural boundary of analyticity The real axis Im z 0 is the natural boundary of the region of analyticity. 09.50.04.0006.01 z z, Series representations Generalized power series Expansions at generic point z z 0 For the function itself z z 0 09.50.06.0003.01 8 K w w 1 w Λ z 0 Λ z 0 1 Λ z 0 3 3 Λ z 0 3 Λ z 0 7 K 1 w K w E w E 1 w K w Λ z 0 1 3 Λ z 0 3 z z 0 8 1 w w Λ z 0 Λ z 0 1 7 K 1 w E w K w E 1 w K w Λ z 0 1 4 Λ z 0 4 E w Λ z 0 Λ z 0 6 7 Λ z 0 5 7 Λ z 0 4 7 Λ z 0 7 Λ z 0 w K w Λ z 0 7 9 Λ z 0 6 13 Λ z 0 5 4 Λ z 0 4 9 Λ z 0 3 11 Λ z 0 18 Λ z 0 w Λ z 0 6 3 Λ z 0 5 Λ z 0 4 Λ z 0 3 9 Λ z 0 10 Λ z 0 3 6 K w 3 z z 0 ; z z 0 w q 1 Π z 0
http://functions.wolfram.com 4 09.50.06.0004.01 z z 0 1 O z z 0 Exponential Fourier series 09.50.06.0001.01 1 178 Π z 744 a k k Π z ; a k Π k 1 1 A k I 1 4 Π k 1 A k h 0 Π h k H, h 1,gcd h, exp ; h H, h mod 1 09.50.06.000.01 1 178 Π z 744 a k k Π z ; a 1 196 884 a 1 493 760 a 3 864 99 970 a 4 0 45 856 56, a 5 333 0 640 600 n 1 a Ν a n 1 1 a n a n a k a n k ; n Ν Ν mod 4 0 4 a Ν a n 3 a a n 1 a n 1 Ν 1 Ν mod 4 1 4 a n 1 1 a n n a Ν a n a k a n k 1 ; n Ν Ν mod 4 4 a Ν a n 4 a a n 1 1 a n 1 Ν 3 Ν mod 4 3 4 n 1 n a n a k a n k 1 k 1 a k a 4 n k a k a 4 n 4 k ; n n 1 a n 1 a k a n k 3 1 k 1 a k a 4 n k a k a 4 n 4 k ; n n n n 1 Differential equations Ordinary nonlinear differential equations 09.50.13.0001.01 36 w z 41 w z 3 w z 4 7 w z 1 w z w 3 z w z 108 w z 1 w z w z 0 ; w z z Transformations Transformations and argument simplifications Argument involving basic arithmetic operations
http://functions.wolfram.com 5 09.50.16.0001.01 z 1 z 09.50.16.000.01 1 z z Identities Functional identities 09.50.17.0001.01 z 1 09.50.17.000.01 1 z 09.50.17.0003.01 64 z 3 110 59 z z 95 3 z z 6000 z 95 3 z 1 510 15 z 187 500 64 z 3 6000 z 187 500 z 1 953 15 0 Differentiation Low-order differentiation 09.50.0.0005.01 0 z z 09.50.0.0001.0 16 K Λ z Λ z Λ z 1 Λ z 1 Λ z Λ z 1 7 Π Λ z 1 Λ z 09.50.0.000.01 z Π z 864 Π z k a k k Π z ; a k Π k 1 1 A k I 1 4 Π k 1 A k h 0 Π h k H, h 1,gcd h, exp ; h H, h mod 1 09.50.0.0003.01 z Π z 864 Π z k a k k Π z ; a k Π k 1 1 A k I 1 4 Π k 1 A k h 0 Π h k H, h 1,gcd h, exp ; h H, h mod 1
http://functions.wolfram.com 6 09.50.0.0004.0 z 64 K Λ z 3 E Λ z Λ z Λ z 1 Λ z 1 Λ z 1 Λ z z 7 Π Λ z 1 Λ z 1 K Λ z Λ z Λ z 1 4 Λ z 6 11 Λ z 5 6 Λ z 4 4 Λ z 3 13 Λ z 18 Λ z 6 09.50.0.0006.01 3 z 18 K Λ z 4 z 3 7 Π 3 Λ z 1 Λ z Λ z 16 Λ z 8 8 Λ z 7 5 Λ z 6 8 Λ z 5 3 Λ z 4 98 Λ z 3 140 Λ z 9 Λ z 3 K Λ z 6 E Λ z Λ z Λ z 1 4 Λ z 6 11 Λ z 5 6 Λ z 4 4 Λ z 3 13 Λ z 18 Λ z 6 K Λ z 3 E Λ z Λ z Λ z 1 Λ z 1 Λ z Λ z 1 09.50.0.0007.01 4 z 51 K Λ z 5 6 Λ z Λ z 1 Λ z 1 z 4 7 Π 4 Λ z 1 Λ z Λ z Λ z 1 E Λ z 3 18 K Λ z Λ z Λ z 1 4 Λ z 6 11 Λ z 5 6 Λ z 4 4 Λ z 3 13 Λ z 18 Λ z 6 E Λ z 6 K Λ z Λ z 16 Λ z 8 8 Λ z 7 5 Λ z 6 8 Λ z 5 3 Λ z 4 98 Λ z 3 140 Λ z 9 Λ z 3 E Λ z K Λ z 3 3 Λ z 10 11 Λ z 9 13 Λ z 8 1 Λ z 7 36 Λ z 6 40 Λ z 5 1380 Λ z 4 340 Λ z 3 175 Λ z 1060 Λ z 1 09.50.0.0008.01 5 z 048 K Λ z 6 15 Λ z Λ z 1 Λ z 1 z 5 7 Π 5 Λ z 1 Λ z Λ z Λ z 1 E Λ z 4 60 K Λ z Λ z Λ z 1 4 Λ z 6 11 Λ z 5 6 Λ z 4 4 Λ z 3 13 Λ z 18 Λ z 6 E Λ z 3 30 K Λ z Λ z 16 Λ z 8 8 Λ z 7 5 Λ z 6 8 Λ z 5 3 Λ z 4 98 Λ z 3 140 Λ z 9 Λ z 3 E Λ z 10 K Λ z 3 3 Λ z 10 11 Λ z 9 13 Λ z 8 1 Λ z 7 36 Λ z 6 40 Λ z 5 1380 Λ z 4 340 Λ z 3 175 Λ z 1060 Λ z 1 E Λ z K Λ z 4 Λ z 64 Λ z 10 64 Λ z 9 0 Λ z 8 149 Λ z 7 131 Λ z 6 96 Λ z 5 3440 Λ z 4 6086 Λ z 3 5774 Λ z 835 Λ z 567 Operations Limit operation 09.50.5.0001.01 lim Ε Ε 0 Representations through equivalent functions With inverse function
http://functions.wolfram.com 7 09.50.7.0001.01 r s z ; r, s 5 r s 1 1 F 1 1, 1 1 ; 1 11 ; 1 Λ, 3 1 7 Λ 1 F 1 1, 7 1 ; 3 ; 1 Λ z 1 1 Re z 0 Λ z With related functions Involving Weierstrass functions Ω 3 Ω 1 09.50.7.000.01 g 3 g 3 7 g 3 ; g, g 3 g Ω 1, Ω 3, g 3 Ω 1, Ω 3 Im Ω 3 Ω 1 0 09.50.7.0003.01 g 3 ; g, g 3 g 1, z, g 3 1, z Im z 0 g 3 7 g 3 Involving theta functions 09.50.7.0004.01 ϑ 0, Π z 8 ϑ 3 0, Π z 8 ϑ 4 0, Π z 8 3 ; Im z 0 54 ϑ 0, Π z ϑ 3 0, Π z ϑ 4 0, Π z 8 Involving other related functions 09.50.7.0005.01 Η z 4 56 Η z 4 3 178 Η z 48 Η z 4 1 z 1 09.50.7.0006.01 178 56 Η z 16 Η z 16 09.50.7.0008.01 178 Η z 8 Η z 8 Η z 16 16 Η z 8 16 Η z Η z 8 09.50.7.0007.01 4 Λ z Λ z 1 3 7 Λ z 1 Λ z 09.50.7.0009.01 3 3 ; Im z 0 Η z 1 7 Η 3 z 1 Η z 1 43 Η 3 z 1 3 178 Η z 36 Η 3 z 1
http://functions.wolfram.com 8 09.50.7.0010.01 Η z 1 3 Η 3 z 1 3 Η z 1 7 Η 3 z 1 3 z 178 Η z 1 Η 3 z 36 09.50.7.0011.01 Η z 1 50 Η 5 z 6 Η z 6 315 Η 5 z 1 3 178 Η z 30 Η 5 z 6 09.50.7.001.01 Η z 1 10 Η 5 z 6 Η z 6 5 Η 5 z 1 3 5 z 178 Η z 6 Η 5 z 30 Zeros 09.50.30.0001.01 Π Π 0 ; z 3 z 3 Theorems Property of modular functions Every modular function m Τ (meaning a function that is invarant under all argument substitutions of the form z a z b c z d where a, b, c, and d are integers and a d b c 1) is a rational function of z. Transcendentality of Klein invariant for algebraic argument The value Α is transcendental for any algebraic Α where Im Α 0 and Α not being a quadratic irrational. Moonshine conecture The dimensions of the representations of the monster group (the largest simple sporadic group) of order 46 3 0 5 9 11 13 3 17 19 3 9 31 41 47 59 71 are simple combinations the Fourier coefficients of 178 z. Solution of Chazy equation The function w z 4 7 Τ Τ 6 Τ 1 Τ Τ Τ 1 Τ Τ is a solution of Chazy equation w z w z w z 3 w z. Replicability of 178 Τ 744 The function 178 Τ 744 is completely replicable: Σ Τ Π Σ 178 1 Π m Σ n Τ c n m ; 178 Τ 744 c n Π Τ. n 1 m 1 n 1
http://functions.wolfram.com 9 History R. Dedekind (1877) F. Klein (1878) R. Fricke (1890 189)
http://functions.wolfram.com 10 Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see http://functions.wolfram.com/notations/. Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: http://functions.wolfram.com/constants/e/ To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: http://functions.wolfram.com/01.03.03.0001.01 This document is currently in a preliminary form. If you have comments or suggestions, please email comments@functions.wolfram.com. 001-008, Wolfram Research, Inc.