Complex Methods | Part IB, 2002

Using contour integration around a rectangle with vertices

x,x,x+iy,x+iy,-x, x, x+i y,-x+i y,

prove that, for all real yy,

+e(x+iy)2dx=+ex2dx\int_{-\infty}^{+\infty} e^{-(x+i y)^{2}} d x=\int_{-\infty}^{+\infty} e^{-x^{2}} d x

Hence derive that the function f(x)=ex2/2f(x)=e^{-x^{2} / 2} is an eigenfunction of the Fourier transform

f^(y)=+f(x)eixydx\widehat{f}(y)=\int_{-\infty}^{+\infty} f(x) e^{-i x y} d x

i.e. f^\widehat{f} is a constant multiple of ff.

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