Difference between revisions of "Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{ | + | $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ |
where $\vartheta_1$ denots the [[Jacobi theta 1]], $\cot$ denotes [[cotangent]], and $\sin$ denotes [[sine]]. | where $\vartheta_1$ denots the [[Jacobi theta 1]], $\cot$ denotes [[cotangent]], and $\sin$ denotes [[sine]]. | ||
Revision as of 07:14, 27 June 2016
Theorem
The following formula holds: $$\dfrac{\vartheta_1'(u,q)}{\vartheta_1(u,q)} = \cot(u)+4\displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ 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) ... (next): 16.29.1
