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Let $f$ and $g$ be two complex valued functions. I want to know whether or not the following two statements are true or not.

  1. $\operatorname{res}(f+g\,; z_0)=\operatorname{res}(f\,; z_0)+\operatorname{res}(g\,; z_0)$
  2. $\operatorname{res}(f \cdot g\,; z_0)=\operatorname{res}(f\,; z_0) \cdot\operatorname{res}(g\,; z_0)$

I think that the first statement is true but the second is false. I'm unsure of how to justify this though. How can I do this?

2 Answers 2

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For the first, you use the fact that the coefficients of the Laurent series are additive (you can use some integral formula and linearity of integral).

For the second, let $f(z) = g(z) = 1/z.$

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If $f(z)=\sum_{=-\infty}^{\infty}a_n(z-z_0)^n$ and $g(z)=\sum_{=-\infty}^{\infty}b_n(z-z_0)^n$, then the residues are $a_{-1}$ and $b_{-1}$

Now, $f+g= \sum_{=-\infty}^{\infty}(a_n+b_n)(z-z_0)^n$, and the first is true

But, in $fg$ you can form "$\frac{1}{z-z_0}$" with $a_{-1}b_0$, $a_{-2}b_1$, etc, ie $res(fg,z_0)=\sum a_{-j}b_{j-1}$ with $j\in \mathbb{Z}$