Logarithm of quotient of Jacobi theta 3 equals a sum of sines

Theorem

The following formula holds: $$\log \left( \dfrac{\vartheta_3(\alpha+\beta,q)}{\vartheta_3(\alpha-\beta,q)} \right)=4\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k}{k} \dfrac{q^k}{1-q^{2k}} \sin(2k\alpha) \sin(2k\beta),$$ where $\log$ denotes the logarithm, $\vartheta_3$ denotes the Jacobi theta 3, and $\sin$ denotes the sine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.30.3$