# Ngô Quốc Anh

## August 17, 2010

### Evaluate complex integral via the Fourier transform

Filed under: Giải tích 7 (MA4247) — Tags: — Ngô Quốc Anh @ 5:56

As suggested from this topic, we are interested in evaluating the following complex integral

$\displaystyle G(t)=\mathop {\lim }\limits_{A \to \infty } \int\limits_{ - A}^A {{{\left( {\frac{{\sin x}} {x}} \right)}^2}{e^{itx}}dx}$.

The trick here is to use the Fourier transform. Thanks to ZY for teaching me this interesting technique.

In $\mathbb R$, the Fourier transform of function $f$, denoted by $\mathcal F[f]$, is defined to be

$\displaystyle \mathcal F[f](y) = \int_{ - \infty }^\infty {f(x){e^{ - 2\pi ixy}}dx}$.

If we apply the Fourier transform twice to a function, we get a spatially reversed version of the function. Precisely,

$\displaystyle\begin{gathered} \mathcal{F}\left[ {\mathcal{F}[f]} \right](z) = \int_{ - \infty }^\infty {\mathcal{F}[f](y){e^{ - 2\pi iyz}}dy} \hfill \\ \qquad\qquad= \int_{ - \infty }^\infty {\mathcal{F}[f](y){e^{2\pi iy( - z)}}dy} \hfill \\ \qquad\qquad= {\mathcal{F}^{ - 1}}\left[ {\mathcal{F}[f]} \right]( - z) \hfill \\ \qquad\qquad= f( - z) \hfill \\ \end{gathered}$

where $\mathcal F^{-1}$ denotes the inverse Fourier transform.

## July 30, 2009

### A couple of complex integrals involving exp(itx) for a real parameter t

In this turn, I will consider a couple of examples of complex contour integrals with respect to variable $x$ involving the following factor $e^{itx}$ where $t$ a real parameter.

Problem 1. Evaluate the integral

$\displaystyle I\left( t \right) = \int\limits_{ - \infty }^\infty {\frac{{{e^{itx}}}} {{{{\left( {x + i} \right)}^2}}}dx}$

where $-\infty < t<\infty$.

Solution. Let

$\displaystyle {f_t}(z) = \frac{{{e^{itz}}}}{{{{(z + i)}^2}}}$

and consider first the case $t>0$. Then $|f_t(z)|$ is bounded in the upper half-plane by

$\displaystyle\frac{1}{|z+i|^2}$.

For $R>1$ let

$\displaystyle C_R=\Gamma_R \cup [-R, R]$,

where $\Gamma_R$ is the semicircle centered at the origin joining $R$ and $-R$, oriented counterclockwise.

## July 17, 2009

### 3 indefinite integral problems involving sinx/x via residue

Problem 1. Compute

$\displaystyle\int\limits_{ - \infty }^\infty {\frac{{\sin x}} {x}dx}$

via complex variable methods.

Problem 2. Compute

$\displaystyle\int\limits_{ - \infty }^\infty {\frac{{\sin^2 x}} {x^2}dx}$

via complex variable methods.

Problem 3. Compute

$\displaystyle\int\limits_{ - \infty }^\infty {\frac{{\sin^3 x}} {x^3}dx}$

via complex variable methods.