The Jacobi theta functions can be expressed as infinite products:

$$\vartheta_1 (z;q) = 2 q^{1/4} \sin z \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos 2 z + q^{4n})$$ $$\vartheta_2 (z;q) = 2 q^{1/4} \cos z \prod_{n=1}^\infty (1 - q^{2n}) (1 + 2 q^{2n} \cos 2 z + q^{4n})$$ $$\vartheta_3 (z;q) = \prod_{n=1}^\infty (1 - q^{2n}) (1 + 2 q^{2n-1} \cos 2 z + q^{4n-2})$$ $$\vartheta_4 (z;q) = \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n-1} \cos 2 z + q^{4n-2})$$