# product representations of Jacobi $\vartheta$ functions

The Jacobi theta functions can be expressed as infinite products:

 $\vartheta_{1}(z;q)=2q^{1/4}\sin z\prod_{n=1}^{\infty}(1-q^{2n})(1-2q^{2n}\cos 2% z+q^{4n})$
 $\vartheta_{2}(z;q)=2q^{1/4}\cos z\prod_{n=1}^{\infty}(1-q^{2n})(1+2q^{2n}\cos 2% z+q^{4n})$
 $\vartheta_{3}(z;q)=\prod_{n=1}^{\infty}(1-q^{2n})(1+2q^{2n-1}\cos 2z+q^{4n-2})$
 $\vartheta_{4}(z;q)=\prod_{n=1}^{\infty}(1-q^{2n})(1-2q^{2n-1}\cos 2z+q^{4n-2})$
Title product representations of Jacobi $\vartheta$ functions ProductRepresentationsOfJacobivarthetaFunctions 2013-03-22 14:52:13 2013-03-22 14:52:13 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Theorem msc 33E05