Squares of theta relation for Jacobi theta 1 and Jacobi theta 4

Theorem

The following formula holds: $$\vartheta_1^2(z,q)\vartheta_4^2(0,q)=\vartheta_3^2(z,q)\vartheta_2^2(0,q)-\vartheta_2^2(z,q)\vartheta_3^2(0,q),$$ where $\vartheta_1$ denotes the Jacobi theta 1, $\vartheta_2$ denotes the Jacobi theta 2, $\vartheta_3$ denotes Jacobi theta 3, and $\vartheta_4$ denotes Jacobi theta 4.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.28.1$