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], use the identities sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A\sin B = \tfrac12[\cos(A-B) - \cos(A+B)] and sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \tfrac12[\sin(A+B) + \sin(A-B)] to prove both ∫−L/2L/2sin⁡2πmxLsin⁡2πnxL dx=L2δmn\int_{-L/2}^{L/2}\sin\frac{2\pi mx}{L}\sin\frac{2\pi nx}{L}\,\mathrm{d}x = \frac{L}{2}\delta_{mn} and ∫−L/2L/2sin⁡2πmxLcos⁡2πnxL dx=0for every m,n.\int_{-L/2}^{L/2}\sin\frac{2\pi mx}{L}\cos\frac{2\pi nx}{L}\,\mathrm{d}x = 0 \quad\text{for every } m, n. (a) Derive the sine-sine relation, treating m≠nm \ne n and m=nm = n separately. (b) Derive the sine-cosine relation, and explain why no separate case for m=nm = n is needed there.

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