EllipticTheta2. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

Transcription

1 EllipticTheta Notations Traditional name Jacobi theta function ϑ Traditional notation ϑ z, Mathematica StandardForm notation EllipticTheta, z, Primary definition ϑ z, cos k 1 z ; 1 Specific values Specialized values For fixed z ϑ z, 0 0 For fixed ϑ 0, Η log log Η ϑ 0, K Η ϑ 0, Τ Τ ; ImΤ 0 ΗΤ

2 ϑ 0, Τ Τ ϑ 0, ; Τ ϑ, ϑ, ϑ 6, Τ 3 Η3 Τ ; ImΤ 0 ReΤ 1 ϑ , Τ 3 Η3 Τ ; ImΤ 0 ReΤ ϑ m, 1m log Η log Η ; m ϑ m 1, 0 ; m ϑ m 1 n Τ, 0 ; m, n Τ General characteristics Domain and analyticity ϑ z, is an analytic function of z and for z, and z ϑ z, Symmetries and periodicities Parity ϑ z, is an even function with respect to z ϑ z, ϑ z, ϑ z, exp sgnim ϑ z,

3 3 Mirror symmetry ϑ z, ϑ z, Periodicity The function ϑ z, is a periodic function with respect to z with period and a uasi-period log ϑ z, ϑ z, ϑ z, ϑ z, ϑ z m, 1 m ϑ z, ; m ϑ z Τ, z ϑ z log, z ϑ z, ; Τ ImΤ 0 ϑ z, ϑ z m Τ, m m zm m1 Τ ϑ z, ; m Τ ϑ z m log, m m z ϑ z, ; m ϑ z m n Τ, 1 m n n z ϑ z, ; m, n Τ Poles and essential singularities With respect to The function ϑ z, does not have poles and essential singularities inside of the unit circle ing ϑ z, With respect to z ing z ϑ z, Branch points With respect to For fixed z, the function ϑ z, has one branch point: 0. (The point 1 is the branch cut endpoint.)

4 ϑ z, ϑ z,, 0 With respect to z For fixed, the function ϑ z, does not have branch points z ϑ z, Branch cuts With respect to For fixed z, the function ϑ z, is a single-valued function inside the unit circle of the complex -plane, cut along the interval 1, 0, where it is continuous from above ϑ z, 1, 0, lim ϑ z, Ε ϑ z, ; 1 0 Ε lim ϑ z, Ε ϑ z, ; 1 0 Ε0 With respect to z For fixed, the function ϑ z, does not have branch cuts z ϑ z, Natural boundary of analyticity The unit circle 1 is the natural boundary of the region of analyticity z ϑ, z, Branch cut endpoints The function ϑ z, has one branch cut endpoint: 1. Series representations -series Expansions at generic point z z 0

5 ϑ z, ϑ z 0, ϑ 1,0 z 0, z z 0 ϑ,0 z 0, ϑ z, ϑ z 0, ϑ z 0, z z 0 ϑ,0 z 0, ϑ k,0 z 0, ϑ z, z z 0 k k ϑ z, ϑ z 0, 1 Oz z 0 z z 0 ϑ 3,0 z 0, 6 z z 0 ϑ 3,0 z 0, 6 z z 0 3 Oz z 0 z z 0 3 Oz z 0 Expansions at generic point ϑ z, ϑ z, 0 ϑ 0,1 z, 0 0 ϑ 0, z, ϑ 0,k z, 0 ϑ z, 0 k k ϑ z, ϑ z, 0 1 O 0 Expansions on branch cuts ϑ z, x 1 x 0 ϑ z, argx ϑ z, x ϑ 0,1 z, x x ϑ 0, z, x argx ϑ 0,k z, x x k ; x 1 x 0 k 0 ϑ 0,3 z, 0 6 x ϑ 0,3 z, x O 0 x 3 O x ; ϑ z, Expansions at 0 argx ϑ z, x 1 O x ; x 1 x ϑ z, cosz cos3 z cos5 z 6 cos7 z 1 ; ϑ z, cos k 1 z ; 1

6 ϑ z, k k1 n1 z k ϑ 0, k k ϑ z, cosz O ; 0 Expansions at ϑ z, 1 3 arg z log 1 log cosh z log log cosh z log ; ϑ z, 1 3 arg1 k 1 k k 1 j j 1 j 0 k j p j,k 1 k z log 1 k 1 k log cosh k z log ; c k 1k k 1 p j,0 1 p j,k 1 k m 1 k j m k m c m p j,km k ϑ z, 1 3 arg1 1 O 1 z log 1 O log cosh z log ; 1 Other -series representations ϑ z, ϑ z, tanz 1 k k sin k z k log ϑ a b, ϑ a b, log cosa b cosa b 1 k k k sin k a sin k b k logϑ z, logϑ 0, log sin z 1 k sin k z k 1 k k ϑ 1 0, ϑ 1 z, ϑ 0, ϑ z, tanz 1 k sin k z ; Imz ImΤ 1 k Τ k

7 ϑ 1 0, ϑ 1 z, ϑ 0, ϑ z, tanz 1 k k sin z ; Imz ImΤ 1 cos z k k Τ Other series representations ϑ z, log z log 1 k 1 k log cosh k z log z ϑ z, exp Τ exp Τ n 1 n z Τ ; Τ ϑ z, Τ 1 n exp n Τ z n ; Τ Product representations ϑ 0, 1 n 1 n n ϑ z, cosz1 k 1 k cos z k Differential euations Partial differential euations The elliptic theta functions satisfy the (one-dimensional) heat euation: ϑ z, ϑ z, ; Τ Τ z ϑ z, ϑ z, z 0 Transformations Transformations and argument simplifications Argument involving basic arithmetic operations

8 8 ϑ z, log z log ϑ z log, log ϑ z Τ, Τ Τ exp z Τ ϑ z, Τ n th root of r 1 n ϑ z, 1n 1 r n r 1 n1 r n1 r log ϑ z, ; n 1 n Multiple angle formulas ϑ n z, n 1 n1 r 1 1 n r 1 r n n1 ϑ z r n 1, ; n r n r ϑ n z, n 1 r n r 1 n1 r n1 ϑ z r, ; n n ϑ n 1 z 1, Τ n1 ϑ n z k Τ 1, n Τ ϑ z 1, Τ ϑ k Τ 1, n Τ n1 ϑk Τ, n Τ n n k z ϑn z k Τ, n Τ ; ImΤ 0 n ϑ n 1 z 1, Τ n1 ϑ n z k Τ 1, n Τ ϑ z 1, Τ ϑ k Τ 1, n Τ n1 ϑk Τ, n Τ n n k z ϑn z k Τ, n Τ ; ImΤ 0 n Identities Functional identities ϑ 0, 9 1 ϑ 0, 3 9 ϑ 0, 3 1 ϑ 0, Differentiation Low-order differentiation

9 9 With respect to z ϑ z, ϑ z, z ϑ z, k k1 k 1 cos k 1 z ; 1 z With respect to ϑ z, ϑ z, ϑ 1 z, ϑ 1 z, ϑ 0, ϑ 3z, ϑ z, ϑ 1 z, ϑ z, 1 ϑ 30, ϑ 0, ϑ 1z, ϑ z, 1 ϑ z, 1 ϑ 30, ϑ 0, Ζ 1; g 1, log, g 3 1, log ϑ z, ϑ z, k k 1 k k1 3 cos k 1 z ; 1 ϑ z, k k1 k k 3 k k 1 cos k 1 z ; 1 Symbolic differentiation With respect to z n ϑ z, k k1 k 1 n cos n k 1 z ; 1 n z n With respect to n ϑ z, 1 n n k k1 k k 1 n 5 n cos k 1 z ; 1 n Fractional integro-differentiation With respect to z Α ϑ z, Α1 z Α With respect to z Α k k1 1F 1; 1 Α, 1 Α ; 1 k 1 z ; 1

10 10 Α ϑ z, Α 1 Α k k1 k k 5 cos k 1 z ; 1 k k Α 5 Integration Indefinite integration Involving only one direct function k k k1 ϑ z, z sin k 1 z ; 1 k 1 Involving only one direct function with respect to k k1 5 cos k 1 z ϑ z, ; 1 k k 1 5 Representations through euivalent functions With related functions Involving theta functions Involving ϑ 1 z, ϑ z, ϑ 1 z, ϑ z, ϑ 1 z, ϑ z, 1 m ϑ 1 m 1 z, ; m Involving ϑ 3 z, ϑ z, z ϑ 3 z Τ Τ, ; ϑ z, m m 1 m1 z ϑ 3 z 1 Τ m 1 Τ, ; m

11 ϑ z, z ϑ 3 z 1 log, ϑ z, m m 1 m1 z ϑ 3 log m 1 z, ; m Involving ϑ z, ϑ z, z ϑ z 1 Τ Τ 1, ; ϑ z, m m 1 m1 z ϑ m 1 1 Τ z, ; m Τ ϑ z, z ϑ z 1 log, ϑ z, m m 1 1 m1 z ϑ m 1 log z, ; m Involving Jacobi functions ϑ z, m ϑ 1 z, m 1 cs Km z 1 m m ϑ z, m ϑ 3 z, m m cd Km z m ϑ z, m ϑ z, m m 1 m cn Km z m Involving Weierstrass functions ϑ z, 1 n n 1 1 n n 1 exp Η 1 Ω 1 z Σ 1 u; g, g 3 ; Ω 1, Ω 3 Ω 1 g, g 3, Ω 3 g, g 3 Η 1 ΖΩ 1 ; g, g 3 exp Ω 3 Ω 1

12 ϑ z, ϑ 0, exp Η 1 Ω 1 z Ω 1 z Σ 1 ; g, g 3 ; Ω 1, Ω 3 Ω 1 g, g 3, Ω 3 g, g 3 Η 1 ΖΩ 1 ; g, g 3 exp Ω 3 Ω ϑ z, ϑ z, Ω 1 Ζ Ω 1 z ; g, g 3 Ω 1Η 1 Ω 1 Η 1 z ; Ω 1, Ω 3 Ω 1 g, g 3, Ω 3 g, g 3 Η 1 ΖΩ 1 ; g, g 3 exp Ω 3 Ω 1 Zeros ϑ z, 0 0 ϑ , 0 ; m ϑ m 1, 0 ; m ϑ m 1 n Τ, 0 ; m, n Τ Theorems The effective potential for the Lagrangian The effective potential VT, Μ, m; A for the Lagrangian ΨΓ Ν Ν i e A Ν Ψ mψψ at temperature T and chemical potential Μ in the one loop approximation is given by VT, Μ, m; A Tr1 1 k B T 0 s ϑ s Μ k B T, s k B T s IAcoths IAsm Μ s, where IA e E B and Tr1 is the corresponding gamma trace. History J. Bernoulli (1713) L. Euler; J. Fourier C. G. J. Jacobi (187) C. W. Borchardt (1838) K. Weierstrass ( )

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