PlanetMath: integral represetations of Jacobi $\vartheta$ functions

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integral represetations of Jacobi $\vartheta$ functions

(Theorem)

The Jacobi theta functions have the following integral representations:

$\displaystyle \vartheta_1 (z \vert \tau) = -e^{iz + i \pi \tau / 4} \int_{i - \... ... + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi \tau u) \over \sin (\pi u)} du$

$\displaystyle \vartheta_2 (z \vert \tau) = -i e^{iz + i \pi \tau / 4} \int_{i -... ...y} {e^{i \pi \tau u^2} \cos (2 u z + \pi u + \pi \tau u) \over \sin (\pi u)} du$

$\displaystyle \vartheta_3 (z \vert \tau) = -i \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi u) \over \sin (\pi u)} du$

$\displaystyle \vartheta_4 (z \vert \tau) = -i \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z) \over \sin (\pi u)} du$

"integral represetations of Jacobi $\vartheta$ functions" is owned by rspuzio. [ full author list (2) ]

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Attachments:: derivation of integral representations of Jacobi $\vartheta$ functions (Derivation) by rspuzio

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Cross-references: representations, integral, Jacobi theta functions
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This is version 9 of integral represetations of Jacobi $\vartheta$ functions, born on 2004-09-30, modified 2008-05-24.
Object id is 6262, canonical name is IntegralRepresetationsOfJacobiVarthetaFunctions.
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AMS MSC:

33E05 (Special functions :: Other special functions :: Elliptic functions and integrals)

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