integral represetations of Jacobi $\vartheta$ functions

The Jacobi theta functions have the following integral representations:

$${\vartheta }_1(z|\tau )=-e^{iz+i\pi \tau /4}{\int }_{i-\mathrm{\infty }}^{i+\mathrm{\infty }}\frac{e^{i\pi \tau u^2}\cos (2uz+\pi \tau u)}{\sin (\pi u)}du$$

$${\vartheta }_2(z|\tau )=-i{e}^{iz+i\pi \tau /4}{\int }_{i-\mathrm{\infty }}^{i+\mathrm{\infty }}\frac{e^{i\pi \tau u^2}\cos (2uz+\pi u+\pi \tau u)}{\sin (\pi u)}du$$

$${\vartheta }_3(z|\tau )=-i{\int }_{i-\mathrm{\infty }}^{i+\mathrm{\infty }}\frac{e^{i\pi \tau u^2}\cos (2uz+\pi u)}{\sin (\pi u)}du$$

$${\vartheta }_4(z|\tau )=-i{\int }_{i-\mathrm{\infty }}^{i+\mathrm{\infty }}\frac{e^{i\pi \tau u^2}\cos (2uz)}{\sin (\pi u)}du$$