7 Fourier Series and Integrals

A Taylor series represents a function by polynomials, and does so well only near a point. A Fourier series represents it by sines and cosines, and does so over a whole interval — including for functions with corners and jumps, which no Taylor series can reach.

The construction rests on orthogonality. Over a period, distinct members of the trigonometric system integrate against one another to zero, so each coefficient can be extracted by integrating the function against the corresponding sine or cosine and the others contribute nothing. That is why the coefficients take the form they do, and it is worth seeing the calculation once rather than accepting the formulae.

The section then treats convergence — what the series converges to at a jump is not obvious and is not always the value of the function — and the economies available when a function is even or odd, where half the coefficients vanish before any integration is done. It closes with functions of arbitrary period and with the passage from series to integrals as the period grows without bound.

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