On the general transformation of the Wirtinger integral (Q2251171)

For the theta functions \(\theta_{00}(\nu,\tau)\), \(\theta_{01}(\nu,\tau)\), \(\theta_{10}(\nu,\tau)\), \(\theta_{11}(\nu,\tau)\), the author considers two functions \(z_1(\tau)\) and \(z_2(\tau)\), called Wirtinger integrals, and discusses the problem to find the coefficients \(A\) and \(B\) with respect to \(\tau\) such that \(z_1\left(\frac{a\tau+b}{c\tau+d}\right)=A\,z_1(\tau)+B\,z_2(\tau)\) holds.