4
$\begingroup$

I'm reading in my notes the proof of Mittag Leffler theorem but when I look at the exercises I don't know how to construct these functions.

From the proof it's clear that if $\{z_n\}$ is the sequence of desired poles. One should construct the meromorphic function $f(z)= \Sigma (R_k - T_k)$ where $R_k$ are the desired principal parts and $T_k$ are rational functions given by Runge's theorem so the sum converges on compact subsets of $\mathbb{C}$

How can i give in a simple case these functions in an explicit form? For example:

A meromorphic function $f(z)$ with simple poles $\forall n \in \mathbb{N}$ with residue equal to $n$

$\endgroup$
6
$\begingroup$

Let's find a meromorphic function $f(z)$ with simple poles $\forall n \in \mathbb{N}$ with residue equal to $n$.

Just follow the proof of Mittag-Leffler's theorem. (I refer to Silverman's Complex Variables)

$P_n \left(\frac{1}{(z-n)}\right) = \frac{n}{z-n}=-\sum_{i=0}^\infty (\frac{z}{n})^i$ where the Maclaurin series converges absolutely in $|z|<n$ and uniformly in $|z|<n/2$

We want to find $Q_n(z)$ such that $\left| P_n \left(\frac{1}{(z-n)}\right) - Q_n(z)\right| < \frac{1}{2^n}$ for $|z|<n/2$, and we'll define $f(z) = \sum_{n=1}^\infty \left[P_n \left(\frac{1}{(z-n)}\right) - Q_n(z)\right]$ as in the proof of Mittag-Leffler's theorem.

$Q_n(z)=-\sum_{i=0}^n (\frac{z}{n})^i$ satisfies the condition. So, we now have the desired function $f(z) = \sum_{n=1}^\infty \left(-\sum_{i=n+1}^\infty (\frac{z}{n})^i\right) = \sum_{n=1}^\infty \frac{z^{n+1}}{n^n (z-n)}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.