The orthogonality relation for complex exponentials is usually stated over

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Fourier series and periodic decomposition

Problem

The orthogonality relation for complex exponentials is usually stated over [−L/2,L/2][-L/2, L/2]. Show that it holds over any interval of length LL: for every real number aa and positive integers m,nm, n, ∫aa+Le2πimx/L e−2πinx/L dx=L δmn,\int_{a}^{a+L} e^{2\pi imx/L}\,e^{-2\pi inx/L}\,\mathrm{d}x = L\,\delta_{mn}, with no dependence of the right-hand side on aa. (a) Evaluate the integral for m=nm = n. (b) Evaluate the integral for m≠nm \ne n by direct integration of the exponential, and show explicitly that the result contains no aa. (c) Confirm that setting a=−L/2a = -L/2 recovers the standard relation.

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