# Ngô Quốc Anh

## 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.