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