Definition:Jacobi Theta Function/First Type
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Definition
The Jacobi Theta function of the first type is defined as:
- $\forall z \in \C: \ds \map {\vartheta_1} {z, q} = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n q^{\paren {n + \frac 1 2}^2} \sin \paren {2 n + 1} z$
where:
- $q = e^{i \pi \tau}$
- $\tau \in \C$ such that $\map \Im \tau > 0$
Also see
- Definition:Jacobi Theta Function of the Second Type
- Definition:Jacobi Theta Function of the Third Type
- Definition:Jacobi Theta Function of the Fourth Type
- Results about the Jacobi theta functions can be found here.
Source of Name
This entry was named for Carl Gustav Jacob Jacobi.
Sources
- 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $21.11$: The four types of Theta-functions
- Weisstein, Eric W. "Jacobi Theta Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiThetaFunctions.html