Jacobi’s identity for $\vartheta$ functions

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.

$${\theta }_1(z\mid -1/\tau )=-i{(-i\tau )}^{1/2}{e}^{\frac{i\tau z^2}{\pi }}{\theta }_1(\tau z\mid \tau )$$

$${\theta }_2(z\mid -1/\tau )={(-i\tau )}^{1/2}{e}^{\frac{i\tau z^2}{\pi }}{\theta }_4(\tau z\mid \tau )$$

$${\theta }_3(z\mid -1/\tau )={(-i\tau )}^{1/2}{e}^{\frac{i\tau z^2}{\pi }}{\theta }_3(\tau z\mid \tau )$$

$${\theta }_4(z\mid -1/\tau )={(-i\tau )}^{1/2}{e}^{\frac{i\tau z^2}{\pi }}{\theta }_2(\tau z\mid \tau )$$

Title	Jacobi’s identity for $\vartheta$ functions
Canonical name	JacobisIdentityForvarthetaFunctions
Date of creation	2013-03-22 14:46:45
Last modified on	2013-03-22 14:46:45
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	5
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 33E05