# Trying to find an elliptic function such that $\lim_{z \rightarrow \lambda} \frac{f'(z)}{f(z)} = x \;\forall \lambda \in \Lambda$

Let $$\Lambda := \{m + in : m,n ∈ \mathbb{Z}\} \subset \mathbb{C}$$ and $$x \in \mathbb{R}$$. I am trying to find an elliptic function $$f : \mathbb{C} \setminus \Lambda \rightarrow \mathbb{C}$$ such that $$\forall \lambda \in \Lambda$$,

$$\lim_{z \rightarrow \lambda} \frac{f'(z)}{f(z)} = x$$

I figured that maybe the Weierstrass elliptic function

$$℘(z,\Lambda) := \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right)$$

could be used but I've tried to write it down and I can't see how I could conclude. Could you please help me?

• Is $x$ the real part of $z$? Or is it a constant? Jan 16 at 17:35
• @user52817: “Let ... and $x \in \Bbb R$” seems pretty clear to me, $x$ is a given real constant. Jan 16 at 17:38
• @Martin R, yes it is clear, just seems like a typo. Jan 16 at 17:44

A (non-constant) elliptic function $$f : \mathbb{C} \setminus \Lambda \to \mathbb{C}$$ has a pole at each $$\lambda \in \Lambda$$, which implies that $$f'/f$$ has a simple pole at all these points. Therefore $$\lim_{z \to \lambda} \frac{f'(z)}{f(z)} = x \in \Bbb R$$ is not possible if $$\lambda \in \Lambda$$.

If you meant $$f$$ is a meromorphic $$\Lambda$$-periodic function,

Then the equation $$x^2=\frac{(\wp_i'(s)^2}{\wp_i(s)^2} = 4\wp_i(s) - g_2(i)/\wp_i(s)$$

gives that $$f(z)=\wp_i(z+s_k)$$ is solution to your problem with $$s_k$$ one of the two solutions of $$\wp_i(s)=\frac{x^2\pm \sqrt{x^4+16 g_2(i)}}{8}$$.

$$\wp_i^{-1}(u)$$ is given by an elliptic integral.

• It is given that $f$ is holomorphic in $\mathbb{C} \setminus \Lambda$, which means that the poles are necessarily in $\Lambda$, so that $f'/f$ cannot have a finite limit at those points. Am I misunderstanding something? Jan 16 at 17:46
• Ah maybe, I interpreted it the opposite way, that is $f$ is a meromorphic function on $\Bbb{C}/\Lambda$. Your interpretation makes the question a nonsense whereas mine makes it interesting. Jan 16 at 17:47
• That's how I interpret “$f : \mathbb{C} \setminus \Lambda \rightarrow \mathbb{C}$” ... Jan 16 at 17:48