# meromorphic extension of a function

Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in \mathbf{Z}$.

We assume that the sum $$f_i(s)=\sum_{\lambda\in \Lambda} \frac{m_{i,\lambda}}{s-\lambda}$$ has only finite non zero terms.

If for $\mathrm{Re}(s)>0$, the sum $\sum_{i=0}^\infty f_i$ absolutely and uniformly converges to a holomorphic function $f$ defined on $\mathrm{Re}(s)>0$.

I would like to know

1. Is $\sum_{i=1}^\infty m_{i,\lambda}$ finite? If so, we call it $m_\lambda$.
2. Does $f$ have a meromorphic extension to $\mathbf{C}$ with pole at $\lambda$ of residue $m_\lambda$.
3. Edit: If we suppose 1, could we get 2?

Thank you!

## 1 Answer

If all $m_{i,\lambda}\geq 0$, the answers to all these questions are "yes". The key observation is that $1/(s-\lambda)$ has positive real part when $s$ is in the right half-plane, while $\lambda$ is in the left half-plane. Therefore even if your series converges at ONE point $s$, then $\Re f_j(s)$ is a series with positive terms, from which all your statements immediately follow. Same if all $m_{i,\lambda}\leq 0$.

EDIT. In general, without positivity assumption, this is not true. The following example was given by user fedja.

For $n\geq 2$, let $p_n$ be rational functions with single (multiple) pole at $-n$, $p_n(z)\to \phi$ uniformly in the right half-plane. Here $\phi$ is an arbitrary bounded analytic function in the half-plane $\Re z>-1$.

Such functions $p_n$ are polynomials of $1/(z+n)$ and they exist by Runge's theorem (in fact one does not need Runge's theorem for approximating on a disk). For each $n$, let $Q_n$ be polynomials, partial sums of $e^z$, of sufficiently high degree $d_n$. The degree is such that the zeros of $Q_n\circ p_n$ are all in the disc $|z+n|<1/2$. This is possible because $Q_n(z)\to e^z$ as $d_n\to\infty$ uniformly on compact subsets of the plane, and $e^z$ has no zeros. (Zeros of $Q_n$ tend to $\infty$ but $p_n$ is close to $\infty$ only near the point $-n$.)

Consider now the rational functions $g_n=Q_n\circ p_n$. They converge to the function $e^\phi$ uniformly in the right half-plane.

Zeros and poles of $g_n$ are in the left half-plane, and they tend to $\infty$ as $n\to\infty$. We take the set of all these zeros and poles as our sequence $\Lambda$.

Finally, $$F_n(z):=g_n'(z)/g_n(z)= \sum_{\lambda\in\Lambda}\frac{m_{\lambda,n}}{z-\lambda},$$ where the sum is finite, and $m_{\lambda,n}$ are integers. This sequence tends to $\phi'$ uniformly in the right half-plane. But $\phi'$ was more or less arbitrary, so you can arrange it to have any singularities in the left half-plane. Now convert the sequence $F_n$ into a series, if you wish.