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The Jacobi theta function for
Definition
For , define the Jacobi theta function by
The terms are real and positive. Since for , the tail is dominated by the geometric series , so the defining series converges absolutely for every .
Depends on
- The complex exponential by its power series
- $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential
- Pi as twice the smallest positive zero of cosine
- The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
- The exponential function is strictly increasing
- The exponential addition formula $\exp(x+y)=\exp(x)\exp(y)$
- For $|r| < 1$, $\sum_{k \ge 0} r^k = 1/(1-r)$, and for $|r| \ge 1$ the series diverges
Used by
Dependency tree · two levels
41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2.1 (standard reference, not scraped)
- K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 12, The Theta Relation (standard reference, not scraped)