Logarithmic derivative of Jacobi theta 3 equals a sum of sines

Theorem

The following formula holds: $$\dfrac{\vartheta_3'(u,q)}{\vartheta_3(u,q)} = 4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^k}{1-q^{2k}} \sin(2ku),$$ where $\vartheta_3$ denotes the Jacobi theta 3 and $\sin$ denotes the sine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.29.3$