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.