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I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2

$$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$

when $m=0$, that to say

- $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
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$$\Theta_2(0,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}$$

I guess that the problem is a simple application of the geometric series, but I do not get the right convergence.

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The problem is not that simple. Well, it’s not very complicated neither, but one needs to use Jacobi’s imaginary transformation. Introducing a new variable $\tau$ by $x=e^{i\pi \tau}$, we obtain

$$\vartheta_2(0|\tau)=\left(-i\tau\right)^{-\frac12}\vartheta_4\left(0|\tau’\right),\qquad \tau’=-\tau^{-1}.$$

Now as $x\to 1$, we have $\tau\to i0$, $\tau’\to i\infty$ and $x’\to 0$, and therefore the asymptotics is given by

$$\vartheta_2\left(0,x\to 1\right)\sim \left(-i\tau\right)^{-\frac12}=\left(-\frac{\ln x}{\pi}\right)^{-\frac12}\sim \left(\frac{1- x}{\pi}\right)^{-\frac12}.$$

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