theorems of analysis  
 
  L'Hospital's rule  
 
  Integration techniques  
 
  Taylor expansions  
 
  Constants  
 
  Units   
 
  home  
 
  ask us  
 


   Complex integration


Contents





© The scientific sentence. 2010

Complex integration :
Infinite integrals
Residu theorem





1. Infinite integrals


To integrate :
\[\Large\bf\color{brown}{ \int_{- \infty}^{+ \infty} \textit f(\textit x) \textit d\textit x } \]
We will use the inegration in the complex plane process.

The associated function f(z) is analytic in the upper half plane (Im (z) ≥ 0); except for a finite number of poles that are not on the real axis (x-axis).

\[\bf\color{black}{ \oint_C \textit f(\textit z) \textit d\textit z = \int_{\Gamma} \textit f(\textit z) \textit d\textit z + \int_{-R}^{+R} \textit f(\textit z) \textit d\textit z } \] \[\bf\color{black}{ R \rightarrow 0 \Rightarrow \int_{\Gamma} \textit f(\textit z) \textit d\textit z = 0 } \] \[ \bf\color{black}{ \oint_C \textit f(\textit z) \textit d\textit z = \int_{-R}^{+R} \textit f(\textit x) \textit d\textit x = 2\pi \textit i \sum_j \textit R_j } \] \[\Large\bf\color{green}{ \int_{- \infty}^{+ \infty} \textit f(\textit x) \textit d\textit x = 2\pi \textit i \sum_j \textit R_j } \]



2. Example


\[ \bf\color{teal}{ \int_0^{+ \infty} \frac{d\textit x}{(x^2 + a^2)^4} \; \; \; \text { a is real.} } \] \[ \bf\color{black}{ \oint_C \frac{d\textit z}{(z^2 + a^2)^4} = \int_{-R}^{+R} \frac{d\textit x}{(x^2 + a^2)^4} + \int_{\Gamma} \frac{d\textit z}{(z^2 + a^2)^4} \\ R \rightarrow 0 \Rightarrow \int_{\Gamma} \frac{d\textit z}{(z^2 + a^2)^4} = 0 \\ \Rightarrow \oint_C \frac{d\textit z}{(z^2 + a^2)^4} = \int_{- \infty}^{+ \infty} \frac{d\textit x}{(x^2 + a^2)^4} } \]

We have two poles zo = - ai and zo = + ai of order 4. Only zo = + ai is concerned. So

\[ \bf\color{black}{ \text {Res(zo = + ai) = } \lim_{\textit z \to zo} \frac{1}{3!} \frac{d^3}{dz^{3}}[(z - ia)^4 \frac{1}{(z^2 + a^2)^4}] = \\ \frac{1}{3!} \frac{d^3}{dz^{3}}[ \frac{1}{(z + ai)^4}] = \frac{-5i}{32 a ^7} } \]

Therefore

\[ \bf\color{black}{ \int_{- \infty}^{+ \infty} \frac{d\textit x}{(x^2 + a^2)^4} = 2\pi \textit i \sum_j \textit R_j = 2\pi \times \frac{-5i}{32 a^7} = \frac{10 \pi}{32 a ^7} \\ \Rightarrow \int_{0}^{+ \infty} \frac{d\textit x}{(x^2 + a^2)^4} = \frac{1}{2} \times \frac{10 \pi}{32 a ^7} = \frac{5 \pi}{32 a ^7} } \]
\[ \Large\bf\color{red}{ \int_{0}^{+ \infty} \frac{d\textit x}{(x^2 + a^2)^4} = \frac{5 \pi}{32 a ^7} } \]






  


chimie labs
|
Physics and Measurements
|
Probability & Statistics
|
Combinatorics - Probability
|
Chimie
|
Optics
|
contact
|


© Scientificsentence 2010. All rights reserved.