JacobiTheta1 - Maple Help

JacobiTheta1

The Jacobi theta function theta1

JacobiTheta2

The Jacobi theta function theta2

JacobiTheta3

The Jacobi theta function theta3

JacobiTheta4

The Jacobi theta function theta4

 Calling Sequence JacobiTheta1(z, q) JacobiTheta2(z, q) JacobiTheta3(z, q) JacobiTheta4(z, q)

Parameters

 z - algebraic expression (the parameter) q - algebraic expression (the nome) such that $\left|q\right|<1$

Description

 • The Jacobi theta functions JacobiTheta1, JacobiTheta2, JacobiTheta3, JacobiTheta4 are defined by:

$\mathrm{JacobiTheta1}\left(z,q\right)=2{q}^{1/4}\left(\sum _{n=0}^{\mathrm{\infty }}{\left(-1\right)}^{n}{q}^{n\left(n+1\right)}\mathrm{sin}\left(\left(2n+1\right)z\right)\right)$

$\mathrm{JacobiTheta2}\left(z,q\right)=2{q}^{1/4}\left(\sum _{n=0}^{\mathrm{\infty }}{q}^{n\left(n+1\right)}\mathrm{cos}\left(\left(2n+1\right)z\right)\right)$

$\mathrm{JacobiTheta3}\left(z,q\right)=1+2\left(\sum _{n=1}^{\mathrm{\infty }}{q}^{{n}^{2}}\mathrm{cos}\left(2nz\right)\right)$

$\mathrm{JacobiTheta4}\left(z,q\right)=1+2\left(\sum _{n=1}^{\mathrm{\infty }}{\left(-1\right)}^{n}{q}^{{n}^{2}}\mathrm{cos}\left(2nz\right)\right)$

 • These series converge very fast for $\left|q\right|<1$. See Chapter 16, "Jacobian Elliptic Functions and Theta Functions" of Handbook of Mathematical Functions edited by Abramowitz and Stegun for more extensive information.

Examples

 > $\mathrm{JacobiTheta1}\left(2.0,0.5\right)$
 ${1.632025903}$ (1)
 > $\mathrm{JacobiTheta2}\left(2.0,0.5\right)$
 ${-0.3181628217}$ (2)
 > $\mathrm{JacobiTheta3}\left(2.0,0.5\right)$
 ${0.3314359783}$ (3)
 > $\mathrm{JacobiTheta4}\left(2.0,0.5\right)$
 ${1.632130562}$ (4)