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

Question

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?

Answer 1

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 $\lambda \in \Lambda$.

Answer 2

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.