Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines

Theorem

The following formula holds: $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ where $\vartheta_1$ denots the Jacobi theta 1, $\cot$ denotes cotangent, and $\sin$ denotes sine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.29.1$