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.

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


$${\theta}_{2}(z\mid 1/\tau )={(i\tau )}^{1/2}{e}^{\frac{i\tau {z}^{2}}{\pi}}{\theta}_{4}(\tau z\mid \tau )$$ 


$${\theta}_{3}(z\mid 1/\tau )={(i\tau )}^{1/2}{e}^{\frac{i\tau {z}^{2}}{\pi}}{\theta}_{3}(\tau z\mid \tau )$$ 


$${\theta}_{4}(z\mid 1/\tau )={(i\tau )}^{1/2}{e}^{\frac{i\tau {z}^{2}}{\pi}}{\theta}_{2}(\tau z\mid \tau )$$ 
