Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines

Theorem

The following formula holds: $$\log \left( \dfrac{\vartheta_2(\alpha+\beta,q)}{\vartheta_2(\alpha-\beta)} \right) = \log \left( \dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)} \right) + 4\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k}{k} \dfrac{q^k}{1-q^{2k}} \sin(2k\alpha)\sin(2k\beta),$$ where $\log$ denotes the logarithm, $\vartheta_2$ denotes the Jacobi theta 2, $\cos$ denotes cosine, and $\sin$ denotee sine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.30.2$