Squares of theta relation for Jacobi theta 3 and Jacobi theta 4

Theorem

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.28.3$