Difference between revisions of "Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines"
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(Created page with "==Theorem== The following formula holds: $$\dfrac{\vartheta_2'(u)}{\vartheta_2(u)} = -\tan(u)+4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ wh...") |
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\dfrac{\vartheta_2'(u)}{\vartheta_2(u)} = -\tan(u)+4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ | + | $$\dfrac{\vartheta_2'(u,q)}{\vartheta_2(u,q)} = -\tan(u)+4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ |
where $\vartheta_2$ denotes the [[Jacobi theta 2]], $\tan$ denotes the [[tangent]], and $\sin$ denotes [[sine]]. | where $\vartheta_2$ denotes the [[Jacobi theta 2]], $\tan$ denotes the [[tangent]], and $\sin$ denotes [[sine]]. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines|next=}}: 16.29.2 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines|next=Logarithmic derivative of Jacobi theta 3 equals a sum of sines}}: $16.29.2$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 18:04, 5 July 2016
Theorem
The following formula holds: $$\dfrac{\vartheta_2'(u,q)}{\vartheta_2(u,q)} = -\tan(u)+4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ where $\vartheta_2$ denotes the Jacobi theta 2, $\tan$ denotes the tangent, and $\sin$ denotes sine.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.29.2$
