Introductions to EllipticThetaPrime4

Transcript

1 Introductions to EllipticThetaPrie Introduction to the Jacobi theta functions General The basic achieveents in studying infinite series were ade in the 18th and 19th centuries when atheaticians investigated issues regarding the convergence of different types of series. In particular, they found that the faous geoetrical series: k converges inside the unit circle z < 1 to the function 1 1, but can be analytically extended outside this circle by the forulas k1 ; z 1 and 0 k ; c k 1 0 k The sus of these two series produce the sae function 1 1. But restrictions on convergence for all three series strongly depend on the distance between the center of expansion 0 and the nearest singular point 1 (where the function 1 1 has a first-order pole). The properties of the series: k lead to siilar results, which attracted the interest of J. Bernoulli (1713), L. Euler, J. Fourier, and other researchers. They found that this series cannot be analytically continued outside the unit circle z < 1 because its boundary z 1 has not one, but an infinite set of dense singular points. This boundary was called the natural boundary of analyticity of the corresponding function, which is defined as the su of the previous series. Special contributions to the theoretical developent of these series were ade by C. G. J. Jacobi (187), who introduced the elliptic aplitude az and studied the twelve elliptic functions cdz, cnz, csz, dcz, dnz, dsz, ncz, ndz, nsz, scz, sdz, snz. All these functions later were naed for Jacobi. C. G. J. Jacobi also introduced four basic theta functions, which can be expressed through the following series:

2 uw, k w k. k These Jacobi elliptic theta functions notated by the sybols ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, have the following representations: ϑ 1 z, 1 k k k1 k1 z k z u z 1, log ϑ z, k k1 n1 z k z u z 1, log ϑ 3 z, n n z u n z log, ϑ z, 1 n n n z u n z,. log A ore detailed theory of elliptic theta functions was developed by C. W. Borchardt (1838), K. Weierstrass ( ), and others. Many relations in the theory of elliptic functions include derivatives of the theta functions with respect to the variable z: ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, which cannot be expressed through other special functions. For this reason, Matheatica includes not only four well-known theta functions, but also their derivatives. Definitions of Jacobi theta functions The Jacobi elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives with respect to z: ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, are defined by the following forulas: ϑ 1 z, 1 k k k1 sin k 1 z ; 1 ϑ z, k k1 cos k 1 z ; 1 ϑ 3 z, k cos k z 1 ; 1 ϑ z, 1 1 k k cos k z ; 1 ϑ 1 z, 1 k k k1 k 1 cos k 1 z ; 1

3 3 ϑ z, k k1 k 1 sin k 1 z ; 1 ϑ 3 z, k k sin k z ; 1 ϑ z, 1 k k k sin k z ; 1. A uick look at the Jacobi theta functions Here is a uick look at the graphics for the Jacobi theta functions along the real axis for 1. ϑ 1 x, 0.5 ϑ x, 0.5 ϑ 3 x, 0.5 ϑ x, 0.5 ϑ 1 x, 0.5 ϑ x, 0.5 ϑ 3 x, 0.5 ϑ x, 0.5 Connections within the group of Jacobi theta functions and with other function groups Representations through related euivalent functions The elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can be represented through the Weierstrass siga functions by the following forulas: ϑ 1 z, Ω 1 exp Η 1 Ω 1 z 1 n 3 Σ Ω 1 z ; 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 ϑ z, 1 n 1 n 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

4 ϑ 3 z, 1 n 1 n1 exp Η 1 Ω 1 z Ω 1 z Σ ; 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 ϑ z, 1 n 1 n1 exp Η 1 Ω 1 z Ω 1 z Σ 3 ; 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, where Ω 1, Ω 3 are the Weierstrass half-periods and Ζz; g, g 3 is the Weierstrass zeta function. The ratios of two different elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can be expressed through corresponding elliptic Jacobi functions with power factors by the following forulas: ϑ 1 z, ϑ z, 1 K z sc 1 1 ϑ 1 z, ϑ 3 z, 1 sd K z ϑ 1 z, ϑ z, sn K z ϑ z, ϑ 1 z, 1 1 cs K z ϑ z, ϑ 3 z, cd K z ϑ z, ϑ z, 1 cn K z ϑ 3 z, ϑ 1 z, 1 K z ds 1 1 ϑ 3 z, ϑ z, 1 dc K z ϑ 3 z, ϑ z, 1 1 dn K z ϑ z, ϑ 1 z, 1 ns K z

5 5 ϑ z, ϑ z, 1 nc K z ϑ z, ϑ 3 z, 1 nd K z, where is an elliptic noe and K is a coplete elliptic integral. Representations through other Jacobi theta functions Each of the theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can be represented through the other theta functions by the following forulas: ϑ 1 z, 1 1 ϑ z 1, ; ϑ 1 z, 1 1 z 1 ϑ 3 z 1 1 log, ; ϑ 1 z, 1 1 z 1 log ϑ z 1, ; ϑ z, 1 ϑ 1 z 1, ; ϑ z, 1 z 1 log ϑ 3 z 1, ; ϑ z, 1 z 1 ϑ z 1 log, ϑ 1 z, 1 z 1 ϑ 1 z 1 1 log, ; ϑ 3 z, 1 z 1 log ϑ z 1, ; ϑ 3 z, ϑ z 1, ; ϑ z, 1 1 z 1 log ϑ 1 z 1, ; ϑ z, 1 z 1 ϑ z 1 log, ϑ z, ϑ 3 z 1, ;.

6 6 The derivatives of the theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can also be expressed through the other theta functions and their derivatives by the following forulas: ϑ 1 z, 1 n n z n ϑ 1 z n log, n ϑ 1 z n log, ;, n ϑ 1 z, 1 1 ϑ z 1 1, ; ϑ 1 z, 1 1 z 1 1 ϑ 3 z 1 1 log, ϑ 3 z 1 1 log, ; ϑ 1 z, 1 1 z 1 1 ϑ z 1 1 log, ϑ z 1 1 log, ; ϑ z, 1 ϑ z, ; ϑ z, 1 n z n n ϑ z n log, ϑ z n log, ;, n ϑ z, 1 z 1 ϑ 3 z 1 1 log, 1 ϑ 3 z 1 1 log, ; ϑ z, 1 z 1 ϑ z 1 1 log, 1 ϑ z 1 1 log, ; ϑ 3 z, 1 z 1 1 ϑ 1 z 1 1 log, ϑ 1 z 1 1 log, ; ϑ 3 z, 1 z 1 1 ϑ z 1 1 log, ϑ z 1 1 log, ; ϑ 3 z, n z n n ϑ 3 z n log, ϑ 3 z n log, ;, n Τ ϑ 3 z, ϑ z 1 1, ; ϑ z, 1 1 z 1 1 ϑ 1 z 1 1 log, ϑ 1 z 1 1 log, ; ϑ z, 1 z 1 1 ϑ z 1 1 log, ϑ z 1 1 log, ; ϑ z, ϑ z, ; ϑ z, 1 n n z n n ϑ z n log, ϑ z n log, ;, n Τ. The best-known properties and forulas for the Jacobi theta functions Values for real arguents

7 7 For real values of the arguents z, (with 1 1), the values of the Jacobi theta functions ϑ 3 z,, ϑ z,, ϑ 3 z,, and ϑ z, are real. For real values of the arguents z, (with 0 1), the values of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 1 z,, and ϑ z, are real. Siple values at zero All Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, have the following siple values at the origin point: ϑ 1 0, 0 0 ϑ 0, 0 0 ϑ 3 0, 0 1 ϑ 0, 0 1 ϑ 1 0, 0 0 ϑ 0, 0 0 ϑ 3 0, 0 0 ϑ 0, 0 0. Specific values for specialized paraeter All Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, have the following siple values if 0: ϑ 1 z, 0 0 ϑ z, 0 0 ϑ 3 z, 0 1 ϑ z, 0 1 ϑ 1 z, 0 0 ϑ z, 0 0 ϑ 3 z, 0 0 ϑ z, 0 0. At the points z 0 and z, all theta functions ϑ 1z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can be expressed through the Dedekind eta function Ηw ; w log or a coposition of the coplete elliptic function K and the inverse elliptic noe K 1 by the following forulas: ϑ 1 0, 0 ϑ 0, log Η Η log ϑ 3 0, K 1 ϑ 0, 1 log Η Η log ϑ 1 0, Η log 3 ϑ 0, 0 ϑ 3 0, 0 ϑ 0, 0 ϑ 1, 1 K 1 ϑ, 0 ϑ 3, ϑ 1, 0 ϑ ϑ log, Η 3 K 1 ϑ, 1 ϑ log Η log Η log, Η 3 log, Η 3. log Η 5 The previous relations can be generalized for the cases z and z, where : ϑ 1, 0 ; ϑ, 1 log Η Η log ;

8 8 ϑ 3, 1 log log Η Η log Η 5 ; ϑ, 1 log Η Η log ; ϑ 1 1, 1 1 K 1 ; ϑ 1, 0 ; ϑ 3 1, 1 1 K 1 ; ϑ 1, 1 log Η log Η log Η 5 ; ϑ 1, 1 Η log 3 ; ϑ, 0 ; ϑ 3, 0 ; ϑ, 0 ; ϑ 1, 0 ; ϑ, 11 Η log 3 ; ϑ 3, 11 Η log 3 ; ϑ, 1 Η log 3 ;. Analyticity All Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, are analytic functions of z and for z, and 1. Poles and essential singularities All Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, do not have poles and essential singularities inside of the unit circle 1. Branch points and branch cuts For fixed z, the functions ϑ 1 z,, ϑ z,, ϑ 1 z,, and ϑ z, have one branch point: 0. (The point 1 is the branch cut endpoint.) For fixed z, the functions ϑ 1 z,, ϑ z,, ϑ 1 z,, and ϑ z, are the single-valued functions inside the unit circle of the coplex -plane, cut along the interval 1, 0, where they are continuous fro above: li Ε0 ϑ 1 z, Ε ϑ 1 z, ; 1 0 li Ε0 ϑ 1 z, Ε ϑ 1 z, ; 1 0 li Ε0 ϑ z, Ε ϑ z, ; 1 0 li Ε0 ϑ z, Ε ϑ z, ; 1 0 li Ε0 ϑ 1 z, Ε ϑ 1 z, ; 1 0 li Ε0 ϑ 1 z, Ε ϑ 1 z, ; 1 0 li Ε0 ϑ z, Ε ϑ z, ; 1 0 li Ε0 ϑ z, Ε ϑ z, ; 1 0. For fixed, the functions ϑ 1 z,, ϑ z,, ϑ 1 z,, and ϑ z, do not have branch points and branch cuts with respect to z. The functions ϑ 3 z,, ϑ z,, ϑ 3 z,, and ϑ z, do not have branch points and branch cuts.

9 9 Natural boundary of analyticity The unit circle 1 is the natural boundary of the region of analyticity for all Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,. Periodicity The Jacobi theta functions ϑ 1 z, and ϑ z, are the periodic functions with respect to z with period and a uasi-period log: z ϑ 1 z, ϑ 1 z, ϑ 1 z log, ϑ 1 z, ϑ z, ϑ z, ϑ z log, z ϑ z,. The Jacobi theta functions ϑ 3 z, and ϑ z, are the periodic functions with respect to z with period and a uasiperiod log: ϑ 3 z, ϑ 3 z, ϑ z, ϑ z, ϑ 3 z log, z ϑ 3 z, z ϑ z log, ϑ z,. The Jacobi theta functions ϑ 1 z, and ϑ z, are the periodic functions with respect to z with period : ϑ 1 z, ϑ 1 z, ϑ z, ϑ z, ϑ 1 z, ϑ 1 z, ϑ z, ϑ z,. The Jacobi theta functions ϑ 3 z, and ϑ z, are the periodic functions with respect to z with period : ϑ 3 z, ϑ 3 z, ϑ z, ϑ z,. The previous forulas are the particular cases of the following general relations that reflect the periodicity and uasi-periodicity of the theta functions by variable z: ϑ 1 z n Τ, 1 n n n z ϑ 1 z, ;, n Τ ϑ z n Τ, 1 n n z ϑ z, ;, n Τ ϑ 3 z n Τ, n n z ϑ 3 z, ;, n Τ ϑ z n Τ, 1 n n n z ϑ z, ;, n Τ ϑ 1 z, 1 ϑ 1 z, ; ϑ z, 1 ϑ z, ; ϑ 3 z, ϑ 3 z, ; ϑ z, ϑ z, ;.

10 10 Parity and syetry All Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, ϑ z,, ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, have irror syetry: ϑ 1 z, ϑ 1 z, ϑ z, ϑ z, ϑ 3 z, ϑ 3 z, ϑ z, ϑ z, ϑ 1 z, ϑ 1 z, ϑ z, ϑ z, ϑ 3 z, ϑ 3 z, ϑ z, ϑ z,. The Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, are odd functions with respect to z: ϑ 1 z, ϑ 1 z, ϑ z, ϑ z, ϑ 3 z, ϑ 3 z, ϑ z, ϑ z,. The other Jacobi theta functions ϑ z,, ϑ 3 z,, ϑ z,, and ϑ 1 z, are even functions with respect to z: ϑ z, ϑ z, ϑ 3 z, ϑ 3 z, ϑ z, ϑ z, ϑ 1 z, ϑ 1 z,. The Jacobi theta functions ϑ 1 z,, ϑ 1 z,, ϑ z,, and ϑ z, satisfy the following parity type relations with respect to : ϑ 1 z, exp sgni ϑ 1z, ϑ z, exp sgni ϑ z, ϑ 1 z, exp Ι sgni ϑ 1 z, ϑ z, exp sgni ϑ z,. The Jacobi theta functions ϑ 3 z,, ϑ 3 z,, ϑ z,, and ϑ z, with arguent can be self-transfored by the following relations: ϑ 3 z, ϑ z, ϑ z, ϑ 3 z, ϑ 3 z, ϑ z, ϑ z, ϑ 3 z,. -series representations All Jacobi elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, have the following series expansions, which can be called -series representations: ϑ 1 z, 1 k k k1 sin k 1 z 1 k k k1 k1 z ; 1 k ϑ z, k k1 cos k 1 z k k1 n1 z ; 1 k ϑ 0, k k1 ϑ 3 z, ϑ 3 0, 1 k cos k z 1 n n z ; 1 n n

11 11 ϑ z, 1 1 k k cos k z 1 n n n z ; 1 n ϑ 0, 1 1 n n ϑ 1 z, 1 k k k1 k 1 cos k 1 z 1 k k 1 k k1 k1 z ; 1 k ϑ 1 0, 1 n n 1 n n1 n 0 ϑ z, k k1 k 1 sin k 1 z k 1 k 1 k1 z ; 1 k ϑ 3 z, k k sin k z k k k z ; 1 k ϑ z, 1 k k k sin k z 1 k k k k z ; 1. k Other series representations The theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, can also be represented through the following series: z ϑ 1 z, exp Τ 1 n exp Τ n 1 n z Τ ; Τ z ϑ z, exp Τ exp Τ n 1 n z Τ ; Τ u ϑ 3 u, exp Τ exp Τ n u n Τ ; Τ z ϑ z, exp Τ 1 n exp Τ n z n Τ ; Τ ϑ 1 z, 3 Τ 3 1 n n z n 1 exp Τ z n 1 ; Τ ϑ z, 3 Τ 3 1 n n z n exp Τ z n ; Τ ϑ 3 z, 3 Τ 3 n n z exp Τ z n ; Τ

12 1 ϑ z, 3 Τ 3 n n z 1 exp Τ z n 1 ; Τ. Product representations The theta functions can be represented through infinite products, for exaple: ϑ 1 z, sinz 1 k 1 k cos z k ϑ 0, 1 n 1 n ϑ z, cosz1 k 1 k cos z k ϑ 3 0, 1 n 1 n1 ϑ 3 z, 1 k 1 k1 cos z k ϑ 0, 1 n 1 n1 ϑ z, 1 j 1 k1 cos z k ϑ 1 0, 3 1 n. Transforations The theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, satisfy nuerous relations that can provide transforations of their arguents, for exaple: ϑ 1 z Τ, Τ Τ Τ exp z Τ Τ ϑ 1z, ; Τ ϑ z Τ, Τ Τ exp z Τ ϑ z, ; Τ

13 13 ϑ 3 z Τ, Τ Τ exp z Τ ϑ 3z, ; Τ ϑ z Τ, Τ Τ exp z Τ ϑ z, ; Τ. Aong those transforations, several kinds can be cobined into specially naed groups: n th root of : r 1 n ϑ j z, 1n 1 r n r 1 n1 k n1 k log ϑ j z, ; n 1 j 1,, 3,. n Multiple angle forulas: ϑ 1 n z, n n n 1 n r 1 r n r 1 n1 ϑ 1 z r, ; n n r 0 ϑ 1 n z, n 1 n n n 1 n r 1 r n r 1 n1 ϑ 1 z r, ; n n r n1 ϑ n z, n 1 n1 r 1 1 n r 1 r n n1 ϑ z r n 1, ; n r 0 1 n r ϑ n z, n 1 r n r 1 1 n r ϑ 3 n z, n 1 r n r 1 1 n r ϑ 3 n z, n 1 r n r 1 1 n r ϑ n z, n 1 r n r 1 1 n r ϑ n z, n 1 r n r 1 n1 r n1 r 0 ϑ z r, ; n n n1 ϑ 3 z r n 1, ; n n1 r n1 ϑ 3 z r, ; n n n1 ϑ z r, ; n n r 0 n1 ϑ z r n, ; n. r n1

14 1 Double-angle forulas (which are not particular cases of the previous group), for exaple: ϑ 1 z, ϑ 1z, ϑ z, ϑ 3 z, ϑ z,. ϑ 0, ϑ 3 0, ϑ 0, ϑ z, ϑ z, ϑ 1 z, ϑ 0, 3. ϑ 3 z, ϑ 3z, ϑ 1 z, ϑ 3 0, 3. ϑ z, ϑ z, ϑ 1 z, ϑ 0, 3. Landen's transforation: ϑ 3 z, ϑ z, ϑ z, ϑ z, ϑ 1 z, ϑ 1 z, Identities ϑ 30, ϑ 0, ϑ 0, ϑ 30, ϑ 0,. ϑ 0, The theta functions at z 0 satisfy nuerous odular identities of the for pϑ 1 0, e 1,1, ϑ 0, e,n 0, where the e i,j are positive integers and p is a ultivariate polynoials over the integers, for exaple: 3 ϑ 0, 9 1 ϑ 0, 3 9 ϑ 0, 3 1 ϑ 0, ϑ 3 0, 3 ϑ 3 0, 9 ϑ 3 0, 1 ϑ 3 0, ϑ 0, 9 1 ϑ 0, 3 9 ϑ 0, 3 1. ϑ 0, Aong the nuerous identities for theta functions, several kinds can be joined into specially naed groups: Relations involving suares:

15 15 ϑ 0, ϑ 3 z, ϑ 0, ϑ 1 z, ϑ 3 0, ϑ z, ϑ z, ϑ 3 0, ϑ 0, ϑ 1 z, ϑ 0, ϑ 3 z, ϑ 0, ϑ z, ϑ 0, ϑ z, ϑ 3 0, ϑ 3 z, ϑ 0, ϑ 1 z, ϑ 0, ϑ 3 z, ϑ 3 0, ϑ z, ϑ 3 0, ϑ 1 z, ϑ 0, ϑ z, ϑ 0, ϑ z,. Relations involving uartic powers: ϑ 0, ϑ 0, ϑ 3 0, ϑ 1 z, ϑ 3 z, ϑ z, ϑ z,. Relations between the four theta functions where the first arguent is zero, for exaple: ϑ 1 0, ϑ 0, ϑ 3 0, ϑ 0,. Addition forulas: ϑ 1 x y, ϑ 1 x y, ϑ 1x, ϑ y, ϑ x, ϑ 1 y, ϑ 0, ϑ 1 x y, ϑ 1 x y, ϑ 1x, ϑ 3 y, ϑ 3 x, ϑ 1 y, ϑ 3 0, ϑ 1 x y, ϑ 1 x y, ϑ 1x, ϑ y, ϑ x, ϑ 1 y, ϑ 0, ϑ x y, ϑ x y, ϑ x, ϑ y, ϑ 1 x, ϑ 1 y, ϑ 0, ϑ x y, ϑ x y, ϑ x, ϑ 3 y, ϑ x, ϑ 1 y, ϑ 3 0, ϑ x y, ϑ x y, ϑ x, ϑ y, ϑ 3 x, ϑ 1 y, ϑ 0, ϑ 3 x y, ϑ 3 x y, ϑ 3x, ϑ y, ϑ x, ϑ 1 y, ϑ 0, ϑ x, ϑ 3 y, ϑ 3 x, ϑ y, ϑ 0, ϑ x, ϑ y, ϑ x, ϑ y, ϑ 3 0, ϑ 3 x, ϑ y, ϑ x, ϑ 3 y, ϑ 0, ϑ 3 x, ϑ 3 y, ϑ x, ϑ y, ϑ 0, ϑ 3 x, ϑ y, ϑ 1 x, ϑ y, ϑ 3 0, ϑ x, ϑ y, ϑ 1 x, ϑ 3 y, ϑ 0, ϑ x, ϑ 3 y, ϑ 1 x, ϑ y, ϑ 0,

16 16 ϑ 3 x y, ϑ 3 x y, ϑ 3x, ϑ 3 y, ϑ 1 x, ϑ 1 y, ϑ x, ϑ y, ϑ x, ϑ y, ϑ 3 0, ϑ 3 0, ϑ 3 x y, ϑ 3 x y, ϑ 3x, ϑ y, ϑ x, ϑ 1 y, ϑ 0, ϑ x y, ϑ x y, ϑ 3x, ϑ 1 y, ϑ x, ϑ y, ϑ 0, ϑ x y, ϑ x y, ϑ x, ϑ 3 y, ϑ x, ϑ 1 y, ϑ 3 0, ϑ x y, ϑ x y, ϑ x, ϑ y, ϑ 1 x, ϑ 1 y, ϑ 0, ϑ x, ϑ 3 y, ϑ 1 x, ϑ y, ϑ 0, ϑ 1 x, ϑ 3 y, ϑ x, ϑ y, ϑ 0, ϑ 1 x, ϑ y, ϑ 3 x, ϑ y, ϑ 3 0, ϑ 3 x, ϑ 3 y, ϑ x, ϑ y,. ϑ 0, Triple addition forulas, for exaple: ϑ 3 x y z, ϑ 3 x, ϑ 3 y, ϑ 3 z, ϑ x y z, ϑ x, ϑ y, ϑ z, ϑ 1 x, ϑ 1 y, ϑ 1 z, ϑ 1 x y z, ϑ x, ϑ y, ϑ z, ϑ x y z, ϑ 1 z, ϑ 3 z, ϑ z, ϑ z,. Representations of derivatives The derivatives of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable z can be expressed by the following forulas: ϑ 1 z, ϑ 1 z, z 1 k k k1 k 1 cos k 1 z ; 1 ϑ z, ϑ z, z k k1 k 1 sin k 1 z ; 1 ϑ 3 z, ϑ 3 z, k k sin k z ; 1 z ϑ z, ϑ z, 1 k k k sin k z ; 1 z ϑ 1 z, z 1 k k k1 k 1 sin k 1 z ; 1 ϑ z, z k k1 k 1 cos k 1 z ; 1 ϑ 3 z, 8 k k cos k z ; 1 z

17 17 ϑ z, 8 1 k1 k k cos k z ; 1. z The derivatives of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable can be expressed by the following forulas: ϑ 1 z, ϑ z, ϑ 1z, ϑ z, 1 k k k 1 k k1 3 sin k 1 z ; 1 k k 1 k k1 3 cos k 1 z ; 1 ϑ 3 z, k1 k cos k z ; 1 ϑ z, 1 k k k1 cos k z ; 1 ϑ 1 z, 3 1 k k k1 k k 1 k 1 cos k 1 z ϑ 1 z, ; 1 ϑ z, ϑ z, 3 ϑ 3 z, k1 k 3 sin k z ; 1 ϑ z, 1 k1 k 3 k1 sin k z ; 1. k k1 k k 1 k 1 sin k 1 z ; 1 The n th -order derivatives of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable z can be expressed by the following forulas: n ϑ 1 z, z n n ϑ z, z n 1 k k k1 k 1 n sin n k 1 z ; 1 n k k1 k 1 n cos n k 1 z ; 1 n n ϑ 3 z, n1 k k n cos n k z ; 1 n z n n ϑ z, n1 1 k k k n cos n k z ; 1 n z n

18 18 n ϑ 1 z, z n 1 k k k1 k 1 n1 cos n k 1 z ; 1 n n ϑ z, z n k k1 k 1 n1 sin n k 1 z ; 1 n n ϑ 3 z, n k k n1 sin n k z ; 1 n z n n ϑ z, n 1 k1 k k n1 sin n z n k z ; 1 n. The n th -order derivatives of Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable can be expressed by the following forulas: n ϑ 1 z, n 1 n 1 k k k1 k k 1 n 5 n sin k 1 z ; 1 n n ϑ z, n 1 n k k1 k k 1 n 5 n cos k 1 z ; 1 n n ϑ 3 z, kn k n 1 n n cos k z ; 1 n n ϑ z, 1 k kn k n 1 n n cos k z ; 1 n n ϑ 1 z, n 1 n ϑ z, n 1 n n 1 k k k1 k 1 k k 1 n 5 n cos k 1 z ; 1 n k k1 k 1 k k 1 n 5 n sin k 1 z ; 1 n n ϑ 3 z, kn k k n 1 n n sin k z ; 1 n n ϑ z, 1 k1 kn k k n 1 n n sin k z ; 1 n. Integration The indefinite integrals of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable z can be expressed by the following forulas: ϑ 1 z, z 1 k k k1 cos k 1 z ; 1 k 1

19 19 ϑ z, z 1 k k k1 sin k 1 z ; 1 k 1 k sin k z ϑ 3 z, z z ; 1 k 1 k k sin k z ϑ z, z z ; 1 k ϑ 1 z, z ϑ 1 z, ϑ z, z ϑ z, ϑ 3 z, z ϑ 3 z, ϑ z, z ϑ z,. The first four sus cannot be expressed in closed for through the naed functions. The indefinite integrals of the Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, with respect to variable can be expressed by the following forulas: 1 k k k1 5 sin k 1 z ϑ 1 z, ; 1 k k 1 5 k k1 5 cos k 1 z ϑ z, ; 1 k k 1 5 k1 cos k z ϑ 3 z, ; 1 k 1 1 k k1 cos k z ϑ z, ; 1 k 1 1 k k1 k 5 ϑ k 1 cos k 1 z 1 z, ; 1 k 1 k 5 k1 k 5 ϑ k 1 sin k 1 z z, ; 1 k 1 k 5

20 0 k k1 sin k z ϑ 3 z, ; 1 k 1 1 k1 k k1 sin k z ϑ z, ; 1. k 1 Partial differential euations The elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, satisfy the one-diensional heat euations: ϑ j z, Τ ϑ j z, Τ ϑ j z, ; Τ j 1,, 3, z ϑ j z, ; Τ j 1,, 3,. z The elliptic theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, satisfy the following second-order partial differential euations: ϑ j z, ϑ j z, 0 ; j 1,, 3, z ϑ j z, ϑ j z, 0 ; j 1,, 3,. z Zeros The Jacobi theta functions ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z,, and their derivatives ϑ 1 z,, ϑ z,, ϑ 3 z,, and ϑ z, are eual to zero in the following points: ϑ 1 z, 0 0 ϑ z, 0 0 ϑ 1 z, 0 0 ϑ z, 0 0 ϑ 3 z, 0 0 ϑ z, 0 0 ϑ 1 n Τ, 0 ;, n Τ ϑ 1 n Τ, 0 ;, n Τ ϑ 3 1 Τ Τ n 1, 0 ;, n ϑ n 1 Τ Τ, 0 ;, n ϑ 1, 0 ; ϑ, 0 ;

21 1 ϑ 3 ϑ, 0 ;, 0 ;. Applications of Jacobi theta functions Applications of the Jacobi theta functions include the analytic solution of the heat euation, suare potential well probles in uantu echanics, Wannier functions in solid state physics, conforal apping of periodic regions, gravitational physics, uantu cosology, coding theory, sphere packings, crystal lattice calculations, and study of the fractional uantu Hall effect.

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