Logarithmic derivative of Jacobi theta 4 equals a sum of sines

Theorem

The following formula holds: $$\dfrac{\vartheta_4'(u,q)}{\vartheta_4(u,q)} = 4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^k}{1-q^{2k}}\sin(2uk),$$ where $\vartheta_4$ denotes the Jacobi theta 4 and $\sin$ denotes the sine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.29.4$