Monday, March 1, 2010

Fourier's great idea

As students enter second year mathematics, they will be introduced to a famous mathematical concept called the "Fourier Series", which (unsurprisingly) developed by Fourier in the early part of the 19th century. The Fourier series is based on a very elegant idea that has proven to be very useful in solving equations described the motion of waves, the flow of heat and almost any function or physical behaviour that has a bit of "up and down" (which mathematicians call "periodic").

The basic idea is that any periodic function can be approximated by combining sine and cos functions in an infinite series:

e.g. f(x) = constant + a1cosx + b1sinx + a2cos2x + b2sin2x ......

In this form, the overall period of this function is 360 degrees (2 pi) - you can easily prove to yourself that when you combine trigonometric functions of different periods, the longest period dominates the overall periodic behaviour of the series. Like the Taylor series (which uses an infinite combination of polynomial terms), the more terms included in the series, the greater convergence between the series and original function.

This idea is, in fact, correct for many continuous and discontinuous functions though Fourier's original development of the series (in 1822) did not elucidate the limits of this theorem. Fourier did develop a very clever way of evaluating the constants in the equation through integrating combinations of f(x) and sine and cosine functions over one period of the function. This procedure, which now can be easily performed by computers ("Mathematica" or my favourite website or by a hard working second year engineering student armed with a table of standard integrals.

The Fourier series, long with Taylor's series, is one of the most important mathematical tools available to engineers and scientists for analysing wave functions (e.g. radio waves, music, surf. etc.), solving differential equations and even as a means for compressing and storing data.

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