PlanetMath: Jacobi's identity for $\vartheta$ functions

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Jacobi's identity for $\vartheta$ functions

(Theorem)

Jacobi's identities describe how theta functions transform under replacing the period with the negative of its reciprocal. Together with the quasiperiodicity relations, they describe the transformations of theta functions under the modular group.

$\displaystyle \theta_1 (z \mid -1/\tau) = -i (-i \tau)^{1/2} e^{i \tau z^2 \over \pi} \theta_1 (\tau z \mid \tau)$

$\displaystyle \theta_2 (z \mid -1/\tau) = (-i \tau)^{1/2} e^{i \tau z^2 \over \pi} \theta_4 (\tau z \mid \tau)$

$\displaystyle \theta_3 (z \mid -1/\tau) = (-i \tau)^{1/2} e^{i \tau z^2 \over \pi} \theta_3 (\tau z \mid \tau)$

$\displaystyle \theta_4 (z \mid -1/\tau) = (-i \tau)^{1/2} e^{i \tau z^2 \over \pi} \theta_2 (\tau z \mid \tau)$

"Jacobi's identity for $\vartheta$ functions" is owned by rspuzio. [ full author list (2) ]

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Attachments:: proof of Jacobi's identity for $\vartheta$ functions (Proof) by rspuzio

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Cross-references: modular group, transformations, relations, quasiperiodicity, reciprocal, negative, period, Transform, functions, Jacobi identities
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This is version 2 of Jacobi's identity for $\vartheta$ functions, born on 2004-10-28, modified 2006-10-03.
Object id is 6427, canonical name is JacobisIdentityForVarthetaFunctions.
Accessed 2379 times total.

Classification:

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

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