Difference between revisions of "Logarithmic derivative of Jacobi theta 3 equals a sum of sines"
From specialfunctionswiki
(Created page with "==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 $\v...") |
(No difference)
|
Revision as of 07:17, 27 June 2016
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.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 16.29.3
