1

If $f(\lambda)$ is meromorphic then is it true that it holds for the residue that $\operatorname{Res}_{\lambda=\lambda_0}(f(\lambda)e^\lambda)=\operatorname{Res}_{\lambda=\lambda_0}(f(\lambda))e^{\lambda_0}$ if $\lambda_0$ is a pole of $f$?

I think this only holds if the pole has order $1.$ Thanks in advance.

Michael Hardy
  • 1
Luke Mathwalker
  • 793
  • 2
    $f(z)=\frac1{z^2}$ has residue $0$ at $0$. $f(z)e^z=e^z/z^2$ has residue $1$ at $0$. So you are right, it does not hold in general. But for poles of order $1$, it should be true for multiplication with any entire function, not just the exponential. – Kusma Jul 29 '18 at 16:56

0 Answers0