Difference between revisions of "Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines"
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithmic derivative of Jacobi theta 4 equals a sum of sines|next=Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines}}: 16.30.1 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithmic derivative of Jacobi theta 4 equals a sum of sines|next=Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines}}: $16.30.1$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 18:05, 5 July 2016
Theorem
The following formula holds: $$\log \left( \dfrac{\vartheta_1(\alpha+\beta,q)}{\vartheta_1(\alpha - \beta,q)} \right)=\log \left( \dfrac{\sin(\alpha+\beta)}{\sin(\alpha-\beta)} \right) +4 \displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2k}}{1-q^{2k}}\sin(2k\alpha)\sin(2k\beta),$$ where $\log$ denotes the logarithm, $\vartheta_1$ denotes the Jacobi theta 1, and
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.30.1$
