Derivative of Jacobi theta 1 at 0

Theorem

The following formula holds: $$\vartheta_1'(0,q)=\vartheta_2(0,q) \vartheta_3(0,q) \vartheta_4(0,q),$$ 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): $16.28.6$