Difference between revisions of "Logarithm of quotient of Jacobi theta 3 equals a sum of sines"
From specialfunctionswiki
(Created page with "==Theorem== The following formula holds: $$\log \left( \dfrac{\vartheta_3(\alpha+\beta,q)}{\vartheta_3(\alpha-\beta,q)} \right)=4\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k...") |
(No difference)
|
Revision as of 16:35, 27 June 2016
Theorem
The following formula holds: $$\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.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 16.30.3
