Difference between revisions of "Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines"
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(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...") |
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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),$$ |
where $\vartheta_1$ denots the [[Jacobi theta 1]], $\cot$ denotes [[cotangent]], and $\sin$ denotes [[sine]]. | where $\vartheta_1$ denots the [[Jacobi theta 1]], $\cot$ denotes [[cotangent]], and $\sin$ denotes [[sine]]. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of Jacobi theta 1 at 0|next=}}: 16.29.1 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of Jacobi theta 1 at 0|next=Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines}}: $16.29.1$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 18:04, 5 July 2016
Theorem
The following formula holds: $$\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): $16.29.1$
