Residue theorem
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Revision 1874
8/21/2026, 1:16:13 PM · Sequoia
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linked exercisesNoneQuelques calculs de résidus
Revision 1854
8/21/2026, 12:46:51 PM · Cypress
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##### Intuition2
The Residue theorem claims that the integral over a closed curve of a [[Fonction Méromorphe| meromorphic function]] only depends on the function [[Pôle d'une fonction méromorphe|pôles] compris 2
The Residue theorem claims that the integral over a closed curve of a [[Fonction Méromorphe| meromorphic function]] only depends on the function [[Pôle d'une fonction méromorphe|poles]] 3
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included in the curve.4
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##### Enoncé du théorème6
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]\!]}$7
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)$8
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.9
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##### Examples :11
$\text{ For:} \\ \gamma: [0,1] \mapsto \mathbb{C}\\ t \mapsto e^{i\pi t}\\\text{- We have: }\int_{\gamma}sin(z)\mathrm{d}z=0 \\ \text{- We have to: }\int_{\gamma}\frac{dz}{z} = 2i\pi$Revision 1853
8/21/2026, 12:44:54 PM · Cypress
Updated text
Compare with revision 18502 changed lines
1
##### Intuition2
The Residue theorem claims that the intégral over a closed curve of a [[Fonction Méromorphe| meromorphic function]] only depends on the function [[Pôle d'une fonction méromorphe|pôles] compris 2
The Residue theorem claims that the integral over a closed curve of a [[Fonction Méromorphe| meromorphic function]] only depends on the function [[Pôle d'une fonction méromorphe|pôles] compris 3
4
5
##### Enoncé du théorème6
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]\!]}$7
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)$8
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.9
10
##### Examples :11
$\text{ For:} \\ \gamma: [0,1] \mapsto \mathbb{C}\\ t \mapsto e^{i\pi t}\\\text{- We have: }\int_{\gamma}sin(z)\mathrm{d}z=0 \\ \text{- We have to: }\int_{\gamma}\frac{dz}{z} = 2i\pi$Revision 1850
8/21/2026, 12:44:02 PM · Cypress
Concept created
##### Intuition
The Residue theorem claims that the intégral over a closed curve of a [[Fonction Méromorphe| meromorphic function]] only depends on the function [[Pôle d'une fonction méromorphe|pôles] compris
##### 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 :
$\text{ For:} \\ \gamma: [0,1] \mapsto \mathbb{C}\\ t \mapsto e^{i\pi t}\\\text{- We have: }\int_{\gamma}sin(z)\mathrm{d}z=0 \\ \text{- We have to: }\int_{\gamma}\frac{dz}{z} = 2i\pi$