product representations of Jacobi $\vartheta$ functions

The Jacobi theta functions can be expressed as infinite products:

$${\vartheta }_1(z;q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n})(1-2q^{2n}\cos 2z+q^{4n})$$

$${\vartheta }_2(z;q)=2q^{1/4}\cos z\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n})(1+2q^{2n}\cos 2z+q^{4n})$$

$${\vartheta }_3(z;q)=\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n})(1+2q^{2n-1}\cos 2z+q^{4n-2})$$

$${\vartheta }_4(z;q)=\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n})(1-2q^{2n-1}\cos 2z+q^{4n-2})$$

Title	product representations of Jacobi $\vartheta$ functions
Canonical name	ProductRepresentationsOfJacobivarthetaFunctions
Date of creation	2013-03-22 14:52:13
Last modified on	2013-03-22 14:52:13
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 33E05