Difference between revisions of "Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines"
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines|next=}}: 16.30.2 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines|next=Logarithm of quotient of Jacobi theta 3 equals a sum of sines}}: $16.30.2$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 18:05, 5 July 2016
Theorem
The following formula holds: $$\log \left( \dfrac{\vartheta_2(\alpha+\beta,q)}{\vartheta_2(\alpha-\beta)} \right) = \log \left( \dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)} \right) + 4\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k}{k} \dfrac{q^k}{1-q^{2k}} \sin(2k\alpha)\sin(2k\beta),$$ where $\log$ denotes the logarithm, $\vartheta_2$ denotes the Jacobi theta 2, $\cos$ denotes cosine, and $\sin$ denotee sine.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.30.2$
