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Unformatted text preview: Chapter 7 Fourier series Syllabus section: 6. Fourier series: full, half and arbitrary range series. Parsevals Theorem. Fourier series can be obtained for any function defined on a finite range, as in the S-L section above. In practice they provide a way to do various calculations with, and to analyse the behaviour of, functions which are periodic, i.e. repeat the same values in a regular pattern. Such a function with period 2 will obey an equation f ( x + 2 ) = f ( x ) . To begin with we will assume = . We know cos nx and sin nx for any integer n have period 2 . So, of course, do the other trigonometric functions such as tan x but these have the disadvantage of becoming unbounded at certain values (for tan x , at x = / 2, for example). Since cos nx and sin nx are the eigenfunctions of an S-L system studied in the previous chapter, we can write f = ( a n cos nx + b n sin nx ) for periodic piecewise differentiable f (in fact, for any function defined on a range of length 2 ). We will modify this way of writing the series slightly soon. Such a series splits f into pieces of different frequency. Examples where this technique (or its general- izations for Fourier transforms, wavelets, etc) is useful and makes sense include: resolution of sound waves into their different harmonics, optics, telecommunications, astronomy, climate variation, water waves, peri- odic behaviour of financial measures, etc. Fouriers original application (in his Theorie analytique de la chaleur in 1822) was related to solutions of the equations governing the propagation of heat in a solid; as an example they can be used to show why cellars maintain an almost constant temperature all year round see the appendix to this chapter. We know from S-L system theory that sin nx and cos nx form a set of orthogonal functions: see (6.23) and (6.25). 76 7.1 Full range Fourier series The idea is to write a function f ( x ) defined for a range of values of x of length 2 , say- x , as a series of trigonometric functions f = 1 2 a + 1 a n cos nx + 1 b n sin nx . (7.1) Here the 1 2 a is really a cos0 x = 1 term (the eigenfunction for = 0): the reason for the half is that integraltext - u 2 d x = integraltext - 1d x = 2 which is double the value of integraltext - u 2 d x for the other values of . There is no point in including a sin0 x = 0 term. Strictly, the use of the equals sign depends on convergence properties which we describe later. From the S-L system results above we have that for all m a m = 1 integraldisplay - f cos mx d x . It was to provide this nice form that we included the 1 2 with the a term: 1 2 a is in fact the average value of f over the range. Similarly, b m = 1 integraldisplay - f sin mx d x ....
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This document was uploaded on 02/10/2010.
- Spring '09