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.

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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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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.

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