A strobe light produces a periodic illuminance on a nearby screen

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 strobe light produces a periodic illuminance I(x)I(x) on a nearby screen, with period LL (xx standing for time). Within one period, −L/2<x≤L/2-L/2 < x \le L/2, it is a single flat-topped pulse of height I0=12.0 luxI_0 = 12.0\,\mathrm{lux} and width L/6L/6, centred on the origin: I(x)={12.0 lux,−L/12<x<L/120,otherwise, for −L/2<x≤L/2I(x) = \begin{cases} 12.0\,\mathrm{lux}, & -L/12 < x < L/12 \\ 0, & \text{otherwise, for } -L/2 < x \le L/2 \end{cases} Using the complex Fourier coefficient cn=1L∫−L/2L/2I(x) e−2πinx/L dxc_n = \dfrac{1}{L}\displaystyle\int_{-L/2}^{L/2} I(x)\,e^{-2\pi i n x/L}\,\mathrm{d}x, find c0c_0, c1c_1 and c2c_2.

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