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 :)

Since
$$
\int_C f(z)\,dz = \int_C \operatorname{Re} f(z)\,dz + i\int_C \operatorname{Im} f(z)\,dz,
$$
you can essentially apply the same criteria in the complex case as in the real case.  Specifically, you just need the real and imaginary parts of the integrands to be single-signed.
Let $Q_j$, $j=1,2,3,4$, denote the four closed quadrants of the complex plane.  If
$$
f_n(z) \in Q_j \quad \text{for all } n \in \mathbb N \text{ and } z \in C
$$
for some $Q_j$ then
$$
\displaystyle \oint_C \sum_n f_n(z)= \sum_n \oint_C f_n(z)
$$
follows from applying the known result for real functions to the real and imaginary parts of the integral.
