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

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Theorem

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

Proof

References

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