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Let $\Theta(z| \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)$ be the Jacobi's theta function, and $$\wp_{\tau}(z)=\frac{1}{z^2}+\sum_{w \in \Lambda^*} \left[\frac{1}{(z+w)^2}-\frac{1}{w^2}\right]$$ be the Weierstrass elliptic function with $\Lambda=\Bbb{Z}+\Bbb{Z}\tau$, $\Lambda^*=\Lambda-0$. I want to show that

Stein, Complex analysis, Ch10 Ex1 $$\frac{(\Theta'(z| \tau))^2-\Theta(z| \tau)\Theta''(z| > \tau)}{\Theta(z| \tau)^2}=\wp(z-1/2-\tau/2) + c_{\tau}$$ where $c_\tau$ can be expressed in terms of the first two derivatives of $\Theta(z| \tau)$ with respect to $z$, at $z=1/2+\tau/2$

I solved it, but I don't know how to represent $c_\tau$. I used Jacobi triple product formula and using identity $$\sum_n \frac{1}{(n+\tau)^2}=\frac{\pi^2}{\sin^2 \pi \tau}$$ I think $$c_\tau=\sum_m\sum_n\frac{1}{(n+m\tau)^2}$$ But how can I represent it as a derivative of theta function at $z=1/2+\tau/2$?

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First of all, there is an error in the edition of Stein and Shakarchi that you are using. In later editions, the first two derivatives of $\Theta(z|\tau)$ has been replaced by the first three derivatives of $\Theta(z|\tau)$.

Here's my attempt to answer your question. I am not sure whether it is satisfactory because the expression of $c_\tau$ also includes three constants, $e_1$, $e_2$, and $e_3$, defined as

$\wp(1/2) = e_1$, $\wp(\tau/2) = e_2$, $\wp(1/2+\tau/2) = e_3$.

For completeness, I will also briefly go through the proof of equality itself.

From Corollary 1.5, we know the LHS, hereafter denoted as $L(z)$, is an elliptic function of order 2 with periods 1 and $\tau$, and with a double pole at $z=1/2+\tau/2$. (From here on I will denote $1/2+\tau/2$ as $z_0$.)

Furthermore, multiplying $L(z)$ by $(z-z_0)^2$ and letting $z\rightarrow z_0$, we can see the coefficient of the double pole $\frac{1}{(z-z_0)^2}$ is exactly 1.

We also know that $\wp(z-z_0)$ is an elliptic function of order 2 with periods 1 and $\tau$, and with a double pole at $z=z_0$. Its coefficient of the double pole is also 1. (This can be easily seen from the definition of $\wp(z)$.)

Therefore, $L(z)-\wp(z)$ is an elliptic function that is entire. It must be a constant. This establishes the desired equality.

To get $c_\tau$, we take the derivative of both sides with respect to $z$, and square both sides. We use Theorem 1.7 from the Chapter 9.

$(L')^2 = (\wp')^2 = 4(\wp - e_1)(\wp - e_2)(\wp - e_3) = 4(L-c_\tau-e_1)(L-c_\tau-e_2)(L-c_\tau-e_3)$.

Setting $z=z_0$, all $\Theta(z|\tau)$ vanish. What is left is an equation of the first three derivatives of $\Theta(z|\tau)$ at $z=z_0$, $c_\tau$, and the three constants $e_1$, $e_2$, and $e_3$.

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