product representations of Jacobi $\vartheta$ functions

The Jacobi theta functions can be expressed as infinite products:

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

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

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

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