Expressing a function in terms of other functions

Question

I would like to know if it is possible to express a smooth function, $f(x)$, in terms of the sum of other functions of the form $$f(x)=\sum_{i=1}^\infty\frac{A_i}{x+c_i},$$ over some finite domain. Where $A_i$ and $c_i$ are arbitrary constants which can be complex. I am only interested in real $x$ and so setting $c_i$ to an imaginary number removes the singularity. I know it is possible to express $f(x)$ in a Fourier series. I am looking for something similar to that.

Answer 1

Any meromorphic function $f(z)$ with poles at $z=c_i$ can be presented as $$ f(z)=\sum_{ij}\frac{A_{ij}}{(z-c_i)^j}. $$

This is the essence of the Mittag-Leffler theorem.