Difference between revisions of "Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines"
From specialfunctionswiki
(Created page with "==Theorem== The following formula holds: $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2n}}{1-q^{2n}} \sin(2nu),$$ where...") |
(No difference)
|
Revision as of 07:12, 27 June 2016
Theorem
The following formula holds: $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2n}}{1-q^{2n}} \sin(2nu),$$ where $\vartheta_1$ denots the Jacobi theta 1, $\cot$ denotes cotangent, and $\sin$ denotes sine.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 16.29.1