Given the following 4 functions: \begin{align} \theta_1(0, \tau) &= 2(q^{1/4} + q^{9/4} + q^{25/4} + \ddots), \\ \theta_2(0, \tau) &= 1-2q+2q^4-2q^9+ \ddots), \\ \theta_3(0, \tau) &= 1+2q+2q^4+2q^9+ \ddots), \\ \theta_4(0, \tau) &= 2\pi(q^{1/4} -3q^{9/4} + 5q^{25/4} - \ddots). \end{align}
Prove that:
$\theta_4(0, \tau) = \pi \bigl(\theta_1(0, \tau) \cdot \theta_2(0, \tau) \cdot\theta_3(0, \tau) \bigr)$
I think it can solved both by calculations or by using some special properties of the theta functions. In this case, note that $\theta_4$ is the derivative of the $\theta$ function as defined in Chandrasekharan's "Elliptic Functions". I would prefer the first method.