Let be fixed and let and be positive integers

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

Let L>0L > 0 be fixed and let mm and nn be positive integers. Working over one period [−L/2,L/2][-L/2, L/2], prove that ∫−L/2L/2cos⁡2πmxLcos⁡2πnxL dx=L2δmn,\int_{-L/2}^{L/2}\cos\frac{2\pi mx}{L}\cos\frac{2\pi nx}{L}\,\mathrm{d}x = \frac{L}{2}\delta_{mn}, where δmn=1\delta_{mn} = 1 if m=nm = n and 00 otherwise. (a) Use the identity cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)] to rewrite the integrand as a sum of two cosines. (b) Evaluate the integral for m≠nm \ne n, justifying why each term vanishes. (c) Evaluate the integral for m=nm = n, treating the two terms separately.

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