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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.

Peanutlex
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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.

user
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  • Thank you for your help. To help me understand would you mind giving an example. What would the expansion for $\cos(x)$ look like? How do you calculate the series for a given $f(x)$? – Peanutlex Mar 20 '20 at 13:36
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    @Peanutlex Cosine is holomorphic function. It means it has no poles. Therefore there is no expansion involving non-constant denominator. But maybe the Weierstrass factorization form will suit your purposes: $$\cos(\pi x)=\prod_{n=1}^\infty\left[1-\left(\frac{2x}{2n-1}\right)^2\right].$$ – user Mar 20 '20 at 13:47
  • Thank you. It seems $$\cos(\pi x)\approx\frac{\sinh(\pi)}{\pi}\sum_{n=-\infty}^{\infty}\frac{\left(-1\right)^{n}}{1+\left(x-n\right)^{2}}.$$ I wonder if we can modify this to get a better approximation. – Peanutlex Mar 20 '20 at 14:11
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    @Peanutlex This is the essence of my answer to your other question. The question you put now is interesting but probably requires more precise definition of the approximation quality and of the allowed set of approximating functions. – user Mar 20 '20 at 14:15