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Let $\theta_1(x,\tau)$ be the first Jacobi's theta function: $$ \theta_1(x)=\theta_1(x,\tau)=-i \sum_{n\in \mathbb Z} (-1)^{n}e^{i \pi (n+1/2)^2\tau}e^{2i\pi(n+1/2)x}\qquad x\in \mathbb C, \, \tau\in \mathbb H=\{ z \, \lvert {\rm Im}(z)>0\, \} $$

Let ' denotes the differentiation with respect to the variable $x$ and let's write $\theta'=\theta_1'(0,\tau), \theta''=\theta_1''(0,\tau)$, etc…

Let's consider the following function of the variable $\tau$: $$ \mu: \mathbb H\rightarrow \mathbb C\, , \; \tau \mapsto \frac{1}{3}\frac{\theta'''}{\theta'}\, . $$

Is this function already known? If yes, some references would be welcome.

Thanks.

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  • $\begingroup$ What "is this indeed the case"?? That the function is well known or what? $\endgroup$ – Timbuc Nov 4 '14 at 16:54
  • $\begingroup$ @Timbuc: question edited. $\endgroup$ – ElBabak Nov 4 '14 at 19:44

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