Elliptic Theta 3

8/13/2019 Elliptic Theta 3

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Introductions to EllipticTheta3

Introduction to the Jacobi theta functions

General

The basic achievements in studying infinite series were made in the 18th and

19th centuries when mathematicians investigated issues regarding the

convergence of different types of series. In particular, they found that the

famous geometrical series:

k0

qk

converges inside the unit circle z< 1 to the function 1 1 q, but can be analytically extended outside this circleby the formulas k0 qk1 ;z 1 and k0 q q0k ;ck1 q0k1 q q01 q0. The sums ofthese two series produce the same function 1 1 q. But restrictions on convergence for all three series stronglydepend on

the distance between the center of expansion q0and the nearest singular point 1 (where the function 1 1 q hasa first-order pole).

The properties of the series:

k0

qk2

lead to similar 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 z1 has not one, but an infinite set of densesingular points. This boundary was called the natural boundary of analyticity

of the corresponding function, which is defined as the sum of the previous

series.

Special contributions to the theoretical development of these series weremade by C. G. J. Jacobi (1827), who introduced the elliptic amplitude amzmand studied the twelve ellipticfunctions cdzm, cnzm, cszm, dczm, dnzm, dszm, nczm, ndzm, nszm, sczm,sdzm, snzm. All these functions later were named for Jacobi. C. G. J. Jacobi alsointroduced four basic theta functions, which can be expressed through the

following series:


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uw, qk

qk2w k.

These Jacobi elliptic theta functions notated by the symbols 1z, q, 2z,q, 3z,q, and 4z,q have thefollowing representations:

1z, q q4 k

1k qkk12 k1 z q4 zu 2 z logq 1,q

2z, q q4 k

qkk12 n1 z q4 z u2 z

logq 1,q

3z, qn

qn2

2 n z u2 z

logq ,q

4z, qn

1n qn2 2 n z u 2 z log

q

,q .

A more detailed theory of elliptic theta functions was developed by C. W.

Borchardt (1838), K. Weierstrass (18621863), and others. Many

relations in the theory of elliptic functions include derivatives of the

theta functions with respect to the variable z: 1 z, q, 2 z,q, 3 z, q, and 4 z, q, which cannot be expressed

through other special functions. For this

reason,Mathematicaincludes not only four well-known theta functions, but also their

derivatives.

Definitions of Jacobi theta functions

The Jacobi elliptic theta functions 1z, q, 2z, q, 3z,q, and 4z, q, and their derivatives with respect to z:1

z,q, 2 z,q, 3 z,q, and 4 z,qare defined by the following formulas:

1z, q 2 q4 k0

1k qkk1 sin2 k 1z ;q 1

2z, q 2 q4 k0

qkk1 cos2 k 1z ;q 1

3z, q 2k1

qk2

cos2 k z 1;q 1

4z, q 1 2k1

1k qk2 cos2 k z ;q 1

1 z, q 2 q4

k0

1k qkk12 k 1 cos2 k 1z ;q 1

http://functions.wolfram.com 2


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2 z, q 2 q4

k0

qkk12 k 1 sin2 k 1z ;q 1

3 z, q 4

k1

qk2

ksin2 k z ;q 1

4 z, q 4

k1

1k k qk2 sin2 k z ;q 1.

A quick look at the Jacobi theta functions

Here is a quick look at the graphics for the Jacobi theta functions along

the real axis for q 1 2.

7.5 5 2.5 0 2.5 5 7.5x

4

2

0

2

4

f

1 x, 0.52 x, 0.53 x, 0.54 x, 0.51

x, 0.52

x, 0.53

x, 0.54

x, 0.5

Connections within the group of Jacobi theta functions and with otherfunction groups

Representations through related equivalent functions

The elliptic theta functions 1z, q, 2z, q, 3z, q, and 4z,qcan be represented through the Weierstrass sigmafunctions by the following

formulas:

1z, q

1q

4 exp 2 11z

2

2n1

1 q2 n3

2 1z

;g2, g3 ;

1,31g2, g3,3g2,g3 1 1;g2, g3 q exp 3

1

2z, q 2 q4

n1

1 q2 n n1

1 q2 n2

exp

2 11z2

21u;g2, g3 ;

1,31g2, g3,3g2,g3 1 1;g2, g3 q exp 3

1



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For real values of the arguments z, q(with 1 q 1), the values of the Jacobi theta functions 3z, q, 4z, q,3

z,q, and 4 z, qare real.For real values of the arguments z, q(with 0 q 1), the values of the Jacobi theta functions 1z,q, 2z, q,1

z,q, and 2 z, qare real.Simple values at zero

All Jacobi theta functions 1z,q, 2z, q, 3z, q, 4z,q, 1 z, q, 2 z, q, 3 z, q, and 4 z,q have thefollowing simple values at the origin point:

10, 0 0 20, 0 0 30, 0 1 40, 0 11

0, 0 0 2 0, 0 0 3 0, 0 0 4 0, 0 0.Specific values for specialized parameter

All Jacobi theta functions 1z,q, 2z, q, 3z, q, 4z,q, 1 z, q, 2 z, q, 3 z, q, and 4 z,q have thefollowing simple values if q 0:

1z, 0 0 2z, 0 0 3z, 0 1 4z, 0 11

z, 0 0 2 z, 0 0 3 z, 0 0 4 z, 0 0.

At the points z 0 and z

2, all theta functions 1z, q, 2z, q, 3z,q, 4z, q, 1 z, q, 2 z, q, 3 z, q, and

4 z,q can be expressed through the Dedekind eta function w ;w logq or a composition of the

complete elliptic function Kand the inverse elliptic nome Kq1qby the following formulas:

10,q 0 20,q 2

logq

2 logq 2

30,q 2 Kq1q 40,q 1

logq

logq2

2

1 0,q 2 logq

3 2 0,q 0

3 0,q 0 4 0,q 0

1 2 , q 2 q1q4 Kq1q 2 2 , q 0

3 2 , q 2 1 q1q4 Kq1q 4 2 , q 1

2 logq

2

logq

2

2 logq

5

1 2 , q 0 2 2 , q 2 logq 3

3

4, q 4 4 logq

3 4 4 , q 4

4 logq 3.

The previous relations can be generalized for the cases z mand z 2 m, where m :

1m , q 0;m 2m , q 21m

logq

2 logq 2 ;m



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3m , q 1

2 logq

2

logq

2

2 logq

5 ;m 4m , q 1

logq

logq2

2 ;m

1 m 12, q1m 2 q1q4 Kq1q ;m 2 m 12,q 0;m

3 m 12, q 2 1 q1q4 Kq1q ;m

4 m 12, q 1

2 logq

2

logq

2

2 logq

5 ;m

1 m, q 2 1m logq

3 ;m 2 m ,q 0;m 3 m2 ,q 0;m 4 m 2 , q 0;m

1 m

2,q 0;m 2 m 2 , q 2 1m1

logq 3 ;m

3 m2 4 , q 4 1m1 4 logq 3 ;m 4 m2 4 , q 4 1m 4 logq 3 ;m .

Analyticity

All Jacobi theta functions 1z, q, 2z, q, 3z,q, 4z, q, 1 z,q, 2 z, q, 3 z, q, and 4 z, q are analyticfunctions of zand qfor z,q and q 1.Poles and essential singularities

All Jacobi theta functions 1z, q, 2z, q, 3z, q, 4z,q, 1 z,q, 2 z,q, 3 z, q, and 4 z, qdo not havepoles and essential singularities inside of the unit circle q 1.Branch points and branch cuts

For fixed z, the functions 1z, q, 2z, q, 1 z, q, and 2 z,qhave one branch point: q 0. (The point q 1 is thebranch cut endpoint.)

For fixed z, the functions 1z,q, 2z,q, 1 z, q, and 2 z,qare the single-valued functions inside the unit circle of thecomplex q-plane, cut along the interval 1, 0, where they are continuous from above:

lim0 1z,q 1z, q ;1 q 0 lim0 1z,q 1z, q ;1 q 0lim0 2z,q 2z, q ;1 q 0 lim0 2z,q 2z, q ;1 q 0lim

0

1

z,q

1

z, q

;1 q 0 lim

0

1

z,q

1

z, q

;1 q 0

lim0 2 z,q 2 z, q ;1 q 0 lim0 2 z,q 2 z, q ;1 q 0.

For fixed q, the functions 1z, q, 2z, q, 1 z, q, and 2 z,qdo not have branch points and branch cuts withrespect to z.

The functions 3z,q, 4z,q, 3 z, q, and 4 z,qdo not have branch points and branch cuts.



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Natural boundary of analyticity

The unit circle q 1 is the natural boundary of the region of analyticity for all Jacobi thetafunctions 1z, q, 2z, q, 3z, q, 4z, q, 1 z, q, 2 z, q, 3 z,q, and 4 z, q.Periodicity

The Jacobi theta functions 1z,qand 2z, qare the periodic functions with respect to zwith period 2and aquasi-period logq:

1z 2, q 1z, q 1z logq, q 2 zq 1z, q2z 2, q 2z, q 2z logq, q 2 zq 2z,q.

The Jacobi theta functions 3z, qand 4z,qare the periodic functions with respect to zwith period and a quasi-period logq:

3z ,q 3z, q 3z logq, q 2 z

q 3z,q4z ,q 4z, q 4z logq, q 2 zq 4z, q.

The Jacobi theta functions 1 z, qand 2 z,qare the periodic functions with respect to zwith period 2:

1 z ,q 1 z, q 1 z 2,q 1 z,q

2 z ,q 2 z, q 2 z 2,q 2 z,q.

The Jacobi theta functions 3 z, qand 4 z, qare the periodic functions with respect to zwith period :

3 z ,q 3 z, q

4

z ,q

4

z, q

.

The previous formulas are the particular cases of the following general

relations that reflect the periodicity and quasi-periodicity of the

theta functions by variable z:

1z m n ,q1mn qn2 2 n z 1z, q ;m,n q

2z m n ,q1m qn2 2 n z 2z, q ;m, n q

3z m n ,qqn2 2 n z 3z, q ;m, n q

4z m n ,q1n

q

n2

2 n z

4z,q ;m, n

q

1 z m ,q1m 1 z, q ;m

2 z m ,q1m 2 z, q ;m

3 z m ,q 3 z, q ;m

4 z m ,q 4 z, q ;m .



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Parity and symmetry

All Jacobi theta functions 1z, q, 2z, q, 3z,q, 4z, q, 1 z, q, 2 z, q, 3 z, q, and 4 z, qhave mirrorsymmetry:

1

z, q

1

z, q

2

z, q

2

z, q

3

z, q

3

z,q

4

z, q

4

z,q

1 z, q 1 z, q 2 z, q 2 z, q 3 z, q 3 z,q 4 z, q 4 z,q.The Jacobi theta functions 1z, q, 2 z, q, 3 z, q, and 4 z, qare odd functions with respect to z:1z,q 1z, q 2 z, q 2 z,q 3 z,q 3 z,q 4 z, q 4 z, q.The other Jacobi theta functions 2z,q, 3z,q, 4z, q, and 1 z,qare even functions with respect to z:2z,q 2z, q 3z, q 3z,q 4z, q 4z,q 1 z,q 1 z, q.The Jacobi theta functions 1z, q, 1 z, q, 2z, q, and 2 z, qsatisfy the following parity type relations withrespect to q:

1z,q exp 4 sgnImq 1z, q 2z,q exp 4 sgnImq 2z, q

1 z,q exp

4sgnImq 1 z, q 2 z,q exp 4 sgnImq 2 z, q.

The Jacobi theta functions 3z,q, 3 z, q, 4z, q, and 4 z, qwith argument qcan be self-transformed by thefollowing relations:

3z,q 4z, q 4z,q 3z,q 3 z, q 4 z,q 4 z,q 3 z, q.q-series representations

All Jacobi elliptic theta functions 1z,q, 2z,q, 3z,q, and 4z, q, and their derivatives 1 z, q, 2 z, q,3

z,q, and 4 z, qhave the following series expansions, which can be called q-series representations:

1z, q 2 q4 k0

1k qkk1 sin2 k 1z q4 k

1k qkk12 k1 z ;q 1

2z, q 2 q4 k0

qkk1 cos2 k 1z q4 k

qkk12 n1 z ;q 1

20,q 2 q4 k0

qkk1

3z, q 2k1

qk2

cos2 k z 1n

qn2

2 n z ;q 1

30,q 1 2n1

qn2



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4 z, q 2

32

32

n

n z

1

2exp

z

n

1

2

2

;q .

Product representations

The theta functions can be represented through infinite products, for

example:

1z, q 2 q4 sinzk1

1 q2 k1 2q2 k cos2z q4 k

20,q 2 q4 n1

1 q2 n1 q2 n2

2

z, q

2 q

4cos

z

k1

1 q2 k

1 2 q2 k cos

2z

q4 k

30,q

n1

1 q2 n 1 q2 n12

3z, q k1

1 q2 k 1 2q2 k1 cos2z q4 k2

40,q n1

1 q2 n 1 q2 n12

4z, qk1

1 q2 j 1 2 q2 k1 cos2z q4 k2

1 0,q 2 q4

n1

1 q2 n3

.

Transformations

The theta functions 1z, q, 2z, q, 3z,q, and 4z,qsatisfy numerous relations that can provide transforma-tions of their

arguments, for example:

1z

,

4

expz2

4 1z, q ;q

2z

,

expz2

4z,q ;q



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Double-angle formulas (which are not particular cases of the previous group),

for example:

12z,q 2 1z,q 2z, q 3z,q 4z, q20,q 30,q 40,q .

22z,q2z, q4 1z, q4

20,q3.

32z,q3z, q4 1z, q4

30,q3.

42z,q 4z, q4

1z, q440,q3

.

Landen's transformation:

3z,q 4z, q42z, q2

30,q 40,q

40,q22z,q 1z, q

12z, q2

30,q 40,q40,q2

.

Identities

The theta functions at z 0 satisfy numerous modular identities of the form p10, qe1,1 ,40,qe4,n 0,where the ei,jare positive integers and pis a multivariate polynomials over the integers, for example:

3 20,q920,q

1

3

9 20,q34

20,q4 1

30,q34

30,q94

30,q30,q9

1

3

1

3 40,q940,q

1

3

9 40,q34

40,q4 1.

Among the numerous identities for theta functions, several kinds can be

oined into specially named groups:

Relations involving squares:



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20,q2 3z,q2 40,q2 1z,q2 30,q2 2z, q2

2z, q2 30,q2 40,q2 1z,q2 20,q2 3z, q2

20,q2 2z,q2 40,q2 4z,q2 30,q2 3z, q2

20,q2 1z,q2 40,q2 3z,q2 30,q2 4z, q2

30,q2 1z,q2 40,q2 2z,q2 20,q2 4z, q2.Relations involving quartic powers:

20,q4 40,q4 30,q4

1z, q4 3z, q4 2z, q4 4z,q4.

Relations between the four theta functions where the first argument is zero,

for example:

1 0,q 20,q 30,q 40,q.

Addition formulas:

1x y,q 1x y, q1x,q2 2y,q2 2x, q2 1y,q2

20,q2

4x,q2 3y, q2 3x, q2 4y,q220,q2

1x y,q 1x y, q1x,q2 3y,q2 3x, q2 1y,q2

3

0,q

2

4x,q2 2y, q2 2x, q2 4y,q2

3

0,q

2

1x y,q 1x y, q1x,q2 4y,q2 4x, q2 1y,q2

40,q2

3x,q2 2y, q2 2x, q2 3y,q240,q2

2x y, q 2x y,q2x, q2 2y, q2 1x,q2 1y, q2

20,q2

3x, q2 3y,q2 4x, q2 4y, q220,q2

2x y,q 2x y, q2x,q2 3y,q2 4x, q2 1y,q2

30,q2

3x,q2 2y, q2 1x, q2 4y,q230,q2

2x y,q 2x y, q 2x,q2

4y,q2

3x, q2

1y,q2

40,q2 4x,q

2

2y, q2

1x, q2

3y,q2

40,q2

3x y, q 3x y,q3x, q2 2y, q2 4x,q2 1y, q2

20,q2

2x, q2 3y,q2 1x, q2 4y, q220,q2



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3x y, q 3x y,q3x, q2 3y, q2 1x,q2 1y, q2

30,q2

2x, q2 2y,q2 4x, q2 4y, q230,q2

3x y, q 3x y,q3x, q2 4y, q2 2x,q2 1y, q2

40,q2

4x, q2 3y,q2 1x, q2 2y, q2

40,q2

4x y, q 4x y,q3x, q2 1y, q2 4x, q2 2y,q2

20,q2

1x,q2 3y, q2 2x,q2 4y, q220,q2

4x y, q 4x y,q4x, q2 3y, q2 2x, q2 1y,q2

30,q2

1x,q2 2y, q2 3x,q2 4y, q230,q2

4x y, q 4x y,q4x, q2 4y, q2 1x, q2 1y,q2

40,q2

3x,q2 3y, q2 2x,q2 2y, q240,q2

.

Triple addition formulas, for example:

3x y z,q 3x, q 3y, q 3z,q 2x y z, q 2x, q 2y, q 2z,q1x,q 1y,q 1z, q 1x y z, q 4x,q 4y,q 4z, q 4x y z, q

1z, q4 3z, q4 2z, q4 4z,q4.Representations of derivatives

The derivatives of the Jacobi theta functions 1z,q, 2z, q, 3z, q, and 4z, q, and their derivatives 1 z,q,2

z,q, 3 z,q, and 4 z,qwith respect to variable zcan be expressed by the following formulas:1z,q

z

1

z, q

2 q

4

k0

1

k qkk1

2 k 1

cos

2 k 1

z

;

q

1

2z,qz

2 z, q 2 q4

k0

qkk12 k 1 sin2 k 1z ;q 1

3z,qz

3 z, q 4

k1

qk2

ksin2 k z ;q 1

4z,qz

4 z, q 4

k1

1k k qk2 sin2 k z ;q 1

1

z,q

z 2 q4

k0

1k qkk12 k 12 sin2 k 1z ;q 1

2 z,qz

2 q4

k0

qkk12 k 12 cos2 k 1z ;q 1

3 z,qz

8k1

qk2

k2 cos2 k z ;q 1



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4 z,qz

8k1

1k1 k2 qk2 cos2 k z ;q 1.

The derivatives of the Jacobi theta functions 1z,q, 2z, q, 3z, q, and 4z, q, and their derivatives 1 z,q,2

z,q, 3 z,q, and 4 z,qwith respect to variable qcan be expressed by the following formulas:1z,q

q

1z,q4 q

2k1

1k kk 1qkk1 34 sin2 k 1z ;q 1

2z,qq

2z,q

4 q 2

k1

kk 1qkk1 34 cos2 k 1z ;q 1

3z,qq

2k1

qk21 k2 cos2 k z ;q 1

4z,qq

2k1

1k k2 qk21 cos2 k z ;q 1

1 z,qq

2 q

3

4 k1

1k qkk1kk 1 2 k 1 cos2 k 1z 1 z, q4 q

;q 1

2 z,qq

2

z,q4 q

2 q

3

4 k1

qkk1 kk 1 2 k 1 sin2 k 1z ;q 1

3 z,qq

4k1

qk21 k3 sin2 k z ;q 1

4

z,q

q 4k1

1k1 k3 qk2

1 sin2 k z ;q 1.

The nth-order derivatives of the Jacobi theta functions 1z, q, 2z, q, 3z, q, and 4z, q, and their derivatives1

z,q, 2 z,q, 3 z,q, and 4 z,qwith respect to variable zcan be expressed by the following formulas:n 1z, q

zn 2 q

4 k0

1kqkk12 k 1n sin n2

2 k 1z;q 1 n

n 2z, qzn

2 q4

k0

qkk12 k 1n cos n2

2 k 1z;q 1 n

n 3z, qzn

2n1 k0

qk2

kn cos n

2 2 k z;q 1 n

n 4z, qzn

2n1 k1

1kqk2 kn cos n2

2 k z;q 1 n



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n 1 z, q

zn 2 q

4 k0

1k qkk12 k 1n1 cos n2

2 k 1z;q 1 n

n 2 z, q

zn 2 q

4 k0

qkk12 k 1n1 sin n2

2 k 1z;q 1 n

n 3 z, q

zn 2n2

k1

qk2

kn1 sin n

2 2 k z;q 1 n

n 4 z, q

zn 2n2

k1

1k1 qk2 kn1 sin n2

2 k z;q 1 n .

The nth-order derivatives of Jacobi theta functions 1z,q, 2z, q, 3z,q, and 4z, q, and their derivatives1

z,q, 2 z,q, 3 z,q, and 4 z,qwith respect to variable qcan be expressed by the following formulas:n 1z, q

qn 2 q

1

4n

k0

1k qkk1 kk 1 n 54 n

sin2 k 1z ;q 1 n

n 2z, qqn

2 q1

4n

k0

qkk1 kk 1 n 54 n

cos2 k 1z ;q 1 n

n 3z, qqn

2k1

qk2n k2 n 1

ncos2 k z ;q 1 n

n 4z, qqn

2k1

1kqk2n k2 n 1n

cos2 k z ;q 1 n

n 1

z, q

qn 2 q1

4n

k0

1k qkk12 k 1 kk 1 n 5

4 ncos2 k 1z ;q 1 n

n 2z, qqn

2 q1

4n

k0

qkk12 k 1 kk 1 n 54 n

sin2 k 1z ;q 1 n

n 3 z, q

qn 4

k1

qk2n kk2 n 1

nsin2 k z ;q 1 n

n 4 z, q

qn 4

k1

1k1 qk2n kk2 n 1n

sin2 k z ;q 1 n .

Integration

The indefinite integrals of the Jacobi theta functions 1z,q, 2z,q, 3z, q, and 4z, q, and their derivatives1

z,q, 2 z,q, 3 z,q, and 4 z,qwith respect to variable zcan be expressed by the following formulas:

1z, qz 2 q4 k0

1k qkk12 k 1

cos2 k 1z ;q 1



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2z, qz 2 q4 k0

1k qkk12 k 1

sin2 k 1z ;q 1

3z, qzz k1

qk2

sin2 k zk

;q 1

4z, qzz k1

1k qk2 sin2 k zk

;q 1

1 z, qz 1z,q

2 z, qz 2z,q

3 z, qz 3z,q

4 z, qz 4z,q.

The first four sums cannot be expressed in closed form through the named

functions.

The indefinite integrals of the Jacobi theta functions 1z,q, 2z,q, 3z, q, and 4z, q, and their derivatives1

z,q, 2 z,q, 3 z,q, and 4 z,qwith respect to variable qcan be expressed by the following formulas:

1z, qq 2k0

1

k q

kk1 54 sin

2 k 1

z

kk 1 54

;q 1

2z, qq 2k0

qkk1 5

4 cos2 k 1zkk 1 5

4

;q 1

3z, qqq 2k1

qk21 cos2 k z

k2 1;q 1

4z, qqq 2k1

1k qk21cos2 k zk2 1

;q 1

1 z, qq 2k0

1k qk1k 542 k 1 cos2 k 1zk 1k 5

4

;q 1

2 z, qq 2k0

qk1k 5

42 k 1 sin2 k 1zk 1k 5

4

;q 1



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3 z, qq 4k1

k qk21 sin2 k z

k2 1;q 1

4 z, qq 4k1

1k1 k qk21 sin2 k zk2 1

;q 1.

Partial differential equations

The elliptic theta functions 1z, q, 2z, q, 3z, q, and 4z, q, and their derivatives 1 z, q, 2 z, q, 3 z, q, and4

z, qsatisfy the one-dimensional heat equations:jz,q

4

2 jz,qz2

;q j 1, 2, 3, 4

j z,q

4

2 j z,q

z2;q j 1, 2, 3, 4.

The elliptic theta functions 1z, q, 2z, q, 3z, q, and 4z, q, and their derivatives 1 z, q, 2 z, q, 3 z, q, and4

z, qsatisfy the following second-order partial differential equations:

4 qjz,q

q

2 jz,qz2

0; j1, 2, 3, 4

4 q

j z,qq

2 j z,q

z2 0; j1, 2, 3, 4.

Zeros

The Jacobi theta functions 1z,q, 2z,q, 3z, q, and 4z, q, and their derivatives 1 z, q, 2 z, q, 3 z,q,and 4 z, qare equal to zero in the following points:1z, 0 0 2z, 0 0 1 z, 0 0 2 z, 0 0 3 z, 0 0 4 z, 0 0

1m n , q 0;m, n q

22 m 1

2 n ,q 0;m,n q

32 m 1

2 2 n 1

2,q 0;m,n q

4 m 2n 1 2

, q 0;m, n q

1

2 m , q 0;m

2 m , q 0;m



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3

m

2,q 0;m

4

m

2,q 0;m .

Applications of Jacobi theta functions

Applications of the Jacobi theta functions include the analytic solution of

the heat equation, square potential well problems in quantum mechanics,

Wannier functions in solid state physics, conformal mapping of periodic

regions, gravitational physics, quantum cosmology, coding theory, sphere

packings, crystal lattice calculations, and study of the fractional quantum

Hall effect.



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Elliptic Theta 3

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