A photodetector records a periodic light-intensity fluctuation

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

A photodetector records a periodic light-intensity fluctuation V(x)V(x), which we expand in the harmonics cos⁡(2πnx/L)\cos(2\pi nx/L) and sin⁡(2πnx/L)\sin(2\pi nx/L) of a stretch of length LL. It is known that V(−x)=−V(x)V(-x) = -V(x) (an odd function of xx) and that the trace already repeats after half of that stretch, V(x+L/2)=V(x)V(x + L/2) = V(x), so its own period is L/2L/2. Using only these two symmetries, and without computing any integral, predict which of the real Fourier coefficients a0a_0, ana_n (n≥1n \ge 1) and bnb_n (n≥1n \ge 1) can be non-zero.

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