= Jacobi derivative formula
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{title2=$\theta_1'(0,\tau)=\pi\theta_2(0,\tau)\theta_3(0,\tau)\theta_4(0,\tau)$}
Use period-one spatial normalization and $Q=e^{\pi i\tau}$. The conventional odd function is $\theta_1=-\theta[1/2,1/2]$ with the characteristic convention of <theta function with characteristics>. Differentiating its <Jacobi triple product> gives $\theta_1'(0)=2\pi Q^{1/4}\prod_{m\geq1}(1-Q^{2m})^3$. The three even constant products have product $2Q^{1/4}\prod_{m\geq1}(1-Q^{2m})^3$, proving the identity. The sign changes if one instead calls the characteristic series itself theta1.
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