When can an infinite sum and complex integral be interchanged? Are there some conditions under which the following two are equal?
$$\displaystyle \oint_C \sum  f_n(z)= \sum \oint_C f_n(z)$$
In the case of real valued functions, the condition $f_n(z) \geq 0$ suffices (see here). Is there a simple condition like that in the case of complex valued analytic functions with simple/double poles?
Thank you :)

Uniform convergence allows you to switch a limit and an integral. So if the $f_n$ converge uniformly to $0$ on $C$ (for instance, if they are the terms of a Laurent series and $C$ is contained inside the annulus of convergence), then $\sum f_n(z)$ converges uniformly on $C$, and it is safe to switch the sum and the integral.
