MMfE - Cauchy Residue Theorem

Cauchy Residue Theorem

Mathematical_Methods_for_Engineering Theorem

$${\int }_{\gamma }f(z)dz=2\pi i\sum _i\mathrm{Res}\{f(z),p_i\}$$

Where:

• $f(z)$ is holomorphic
• $\gamma$ is a simple closed positively oriented contour
• $p_i$ are the poles of $f(z)$

NOTE:

• If the curve $\gamma$ is defined in the ANTI-CLOCK wise $\circlearrowleft$ sense then its $+2\pi i$.
• If the curve $\gamma$ is defined in the CLOCK wise $\circlearrowright$ sense then its $-2\pi i$.

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Residues:

$Res\{f(z),p_i\}$ is calculated as follows:

• If $p_i$ is outside the closed contour $\gamma$ :

$$Res\{f(z),p_i\}=0$$

• If $p_i$ is inside $\gamma$, and is a simple pole (multiplicity = 1):

$$Res\{f(z),p_i\}=\underset{z\to p_i}{\lim }(z-p_i)f(z)$$

• If $p_i$ is inside $\gamma$, and is a complex pole of multiplicity = $k$ :

$$Res\{f(z),p_i\}=\frac{1}{(k-1)!}\underset{z\to p_i}{\lim }\frac{d^{k-1}}{d{z}^{k-1}}\left\{(z-p_i{)}^{k}f(z)\right\}$$
- [[MMfE - Holomorphic or Analytical Functions|What's an Holomorphic Function?]] - [[MMfE - Simple Closed Positively Oriented Contour|What's a Simple Closed Positively Oriented Contour?]] - [[MMfE - Poles of a Function|What are the Poles of a Function?]] - [[MMfE - Multiplicity|What's the Multiplicity?]]

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Original File:

Pasted image 20220604185021.png IL LIBRO HA SBAGLIATO, C’È UN ERRORE NELL’ULTIMA FORMA! Integrals of holomorphic functions_220518_145829.pdf cauchy residue theorem_220518_160518.pdf