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.

integrationofexpitx1

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