I've been assigned two integrals to calculate in Fourier Analysis: $$\int_{-\infty}^{\infty}\left(\frac{\sin x}{x}\right)^2dx$$ $$\int_{-\infty}^{\infty}\left(\frac{1-\cos(\lambda\pi)}{\lambda^2}\right)^2d\lambda$$ But I'm not at all sure of how to get started. There have been given hints as to look at the rectangular wave function for the first integral and the triangular wave for the second.
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1Hint: $1-\cos{x} = 2 \sin^2{(x/2)}$ – Ron Gordon Sep 29 '14 at 21:05
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Thanks, but I really would like an elaborate answer instead of just new hints :) – Chris Sep 29 '14 at 21:15
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http://math.stackexchange.com/questions/106570/how-do-i-show-that-int-infty-infty-frac-sin-x-sin-nxx2-dx-pi – ClassicStyle Sep 29 '14 at 23:14
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Never mind, I figured it out myself. Parseval's Theorem says that the Fourier transform preserves the $L^2$ inner product which means that: $$\int_{-\infty}^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\pi$$ $$\int_{-\infty}^{\infty}\left(\frac{1-\cos (\lambda\pi)}{\lambda^2}\right)^2d\lambda=\frac{\pi^4}{3}$$
Chris
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