Residue theorem

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Intuition

The Residue theorem claims that the integral over a closed curve of a meromorphic function^FR only depends on the function poles
included in the curve.

Enoncé du théorème

Let $f$ be a meromorphic function over a simply connected open $U\subset \mathbb{C}$ with n isolated pole(s): the $(p_{i})_{i\in [\![1,n]\!]}$
Then we define $\gamma :[0,1] \mapsto \mathbb{C}$ a cloed continuous closed curve[ $\left(\gamma(0)=\gamma(1) \right)$ ]. Then, we have : $\int_{\gamma}f=\sum_{k=1}^n2i\pi.Ind(\gamma,p_{k}).Res(f,p_k)$
With $Ind(\gamma,p)$ the number of ... $\gamma$ does around de p et $Res(f,p)$ the coefficient $a_{-1}$ of $\frac{1}{z-p}$ in the Laurent serie expansion of the function f around c.

Examples :

$\text{ For:} \\ \gamma: [0,1] \mapsto \mathbb{C}\\ t \mapsto e^{i\pi t}\\\text{- We have: }\int_{\gamma}sin(z)\mathrm{d}z=0 \\ \text{- We have to: }\int_{\gamma}\frac{dz}{z} = 2i\pi$

Practice this concept with exercises

Déterminer la valeur des intégrales suivantes:
1. Pour $\gamma\colon t\in [0,1]\longmapsto e^{2i\pi t}$ , $\displaystyle\int_{\gamma}\frac{e^z}{z}\,\mathrm{d}z$ .
2. Pour $\gamma\colon t\in [0,1]\longmapsto 1+2e^{4i\pi t}$ , $\displaystyle\int_{\gamma}\frac{\cos(z)}{z}\,\mathrm{d}z$ .
3. Pour $\gamma\colon t\in [0,1]\longmapsto e^{-2i\pi t}$ , $\displaystyle\int_{\gamma}\frac{1}{\sin(z)}\,\mathrm{d}z$ .
4. Pour $\gamma$ le rectangle, considéré dans le sens anti-horaire, de sommets $0, 2, 2+2i$ et $2i$ , $\displaystyle\int_{\gamma}\frac{z}{(z-1)^2}\,\mathrm{d}z$ .
Open exercise

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