Blericker74

2022-07-16

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 $\overline{\mathbb{C}}$. we have to show that f can be written as $f={f}_{1}+\dots ,{f}_{p}$ where each ${f}_{j}$ is a rational function.

Keegan Barry

Beginner2022-07-17Added 18 answers

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.

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