Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Transcription

1 KleinInvariant Notations Traditional name Klein invariant modular function Traditional notation z Mathematica StandardForm notation KleinInvariant z Primary definition 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 m 1 ; m Values at fixed points Values at complex z Values at quadratic irrationalities (Heegner Numbers)

2 Values at infinities General characteristics Domain and analyticity z is an analytical function of z which is defined over the upper half of the complex z-plane z z Symmetries and periodicities Periodicity

3 m z ; m Poles and essential singularities On the boundary of analyticity the function z has a dense set of poles ing z z ; Im z 0 Branch points The function z does not have branch points z z Branch cuts The function z does not have branch cuts z z Natural boundary of analyticity The real axis Im z 0 is the natural boundary of the region of analyticity z z, Series representations Generalized power series Expansions at generic point z z 0 For the function itself z z K w w 1 w Λ z 0 Λ z 0 1 Λ z Λ z 0 3 Λ z 0 7 K 1 w K w E w E 1 w K w Λ z Λ z 0 3 z z w w Λ z 0 Λ z K 1 w E w K w E 1 w K w Λ z Λ z 0 4 E w Λ z 0 Λ z Λ z Λ z Λ z 0 7 Λ z 0 w K w Λ z Λ z Λ z Λ z Λ z Λ z 0 18 Λ z 0 w Λ z Λ z 0 5 Λ z 0 4 Λ z Λ z 0 10 Λ z K w 3 z z 0 ; z z 0 w q 1 Π z 0

4 z z 0 1 O z z 0 Exponential Fourier series 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 Π z 744 a k k Π z ; a a a a , a n 1 a Ν a n 1 1 a n a n a k a n k ; n Ν Ν mod a Ν a n 3 a a n 1 a n 1 Ν 1 Ν mod 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 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 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

5 z 1 z z z Identities Functional identities z z z z z 95 3 z z 6000 z 95 3 z z z z z Differentiation Low-order differentiation z z K Λ z Λ z Λ z 1 Λ z 1 Λ z Λ z 1 7 Π Λ z 1 Λ z 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 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

6 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 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 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 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 Λ z 9 Λ z 3 E Λ z K Λ z 3 3 Λ z Λ z 9 13 Λ z 8 1 Λ z 7 36 Λ z 6 40 Λ z Λ z Λ z Λ z 1060 Λ z 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 Λ z 9 Λ z 3 E Λ z 10 K Λ z 3 3 Λ z Λ z 9 13 Λ z 8 1 Λ z 7 36 Λ z 6 40 Λ z Λ z Λ z Λ z 1060 Λ z 1 E Λ z K Λ z 4 Λ z 64 Λ z Λ z 9 0 Λ z Λ z Λ z 6 96 Λ z Λ z Λ z Λ z 835 Λ z 567 Operations Limit operation lim Ε Ε 0 Representations through equivalent functions With inverse function

7 r s z ; r, s 5 r s 1 1 F 1 1, 1 1 ; 1 11 ; 1 Λ, 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 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 g 3 ; g, g 3 g 1, z, g 3 1, z Im z 0 g 3 7 g 3 Involving theta functions 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 Η z 4 56 Η z Η z 48 Η z 4 1 z Η z 16 Η z Η z 8 Η z 8 Η z Η z 8 16 Η z Η z Λ z Λ z Λ z 1 Λ z ; Im z 0 Η z 1 7 Η 3 z 1 Η z 1 43 Η 3 z Η z 36 Η 3 z 1

8 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 Η z 1 50 Η 5 z 6 Η z Η 5 z Η z 30 Η 5 z Η z 1 10 Η 5 z 6 Η z 6 5 Η 5 z z 178 Η z 6 Η 5 z 30 Zeros 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 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 Π m Σ n Τ c n m ; 178 Τ 744 c n Π Τ. n 1 m 1 n 1

9 9 History R. Dedekind (1877) F. Klein (1878) R. Fricke ( )

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