# decomposition into the sum of rational functions

could any one give me a hint for this one? please not the whole solution Let $f$ be a non constant rational function and $z_1,\dots, z_n$ be its poles in $\bar{\mathbb{C}}$. we have to show that $f$ can be written as $f=f_1+\dots,f_p$ where each $f_j$ is a rational function.

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and is there supposed to be some relation between $p$ and $n$, and between the $f_j$ and the $z_i$? –  Gerry Myerson Jun 24 '12 at 7:52
I would guess that each $f_j$ is supposed to have only one pole. In that case the magic phrase is either "partial fractions" or "principal parts". –  Robert Israel Jun 24 '12 at 8:18

Under Robert Israel's interpretation, you can do induction on the number of poles. Expand $f$ in Laurent series in $\{z:0<|z-z_n|<r\}$ for sufficiently small $r$. The series has finitely many negative powers; puts them into $f_n$. Apply the inductive hypotheses to $f-f_n$ and conclude.