Mathematical Methods in the Physical Sciences · Problem 10.20

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · miscellaneous-fourier-transform-parseval

Problem

Find the exponential Fourier transform of

f(x) = \begin{cases} 2a - |x|, & |x| < 2a, \\ 0, & |x| > 2a, \end{cases}

and use your result with Parseval's theorem to evaluate

\int_0^{\infty} \frac{\sin^4 a\alpha}{\alpha^4}\,d\alpha.