proof of Jacobi’s identity for ϑ functions

We start with the Fourier transformDlmfMathworldPlanetmath of f(x)=eiπτx2+2ixz:


Applying the Poisson summation formula, we obtain the following:


The left hand equals ϑ3(zτ). The right hand can be rewritten as follows:


Combining the two expressions yields

Title proof of Jacobi’s identity for ϑ functionsMathworldPlanetmath
Canonical name ProofOfJacobisIdentityForvarthetaFunctions
Date of creation 2013-03-22 14:47:01
Last modified on 2013-03-22 14:47:01
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 19
Author rspuzio (6075)
Entry type Proof
Classification msc 33E05