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fourier series

# fourier series - Chapter 7 Fourier series Syllabus section...

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Chapter 7 Fourier series Syllabus section: 6. Fourier series: full, half and arbitrary range series. Parseval’s 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 = π . We know cos nx and sin nx for any integer n have period 2 π . 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 = π / 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 = 0 ( 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 π ). 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. Fourier’s 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

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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 π , say - π x π , as a series of trigonometric functions f = 1 2 a 0 + 1 a n cos nx + 1 b n sin nx . (7.1) Here the 1 2 a 0 is really a cos0 x = 1 term (the eigenfunction for λ = 0): the reason for the half is that integraltext π - π u 2 0 d x = integraltext π - π 1d x = 2 π which is double the value of integraltext π - π u 2 λ d x for the other values of λ . 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 0 a m = 1 π integraldisplay π - π f cos mx d x . It was to provide this nice form that we included the 1 2 with the a 0 term: 1 2 a 0 is in fact the average value of f over the range. Similarly, b m = 1 π integraldisplay π - π f sin mx d x . Example 7.1. Find the Fourier series for f ( x ) = braceleftBig 0 if - π < x < 0 x if 0 < x < π .
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fourier series - Chapter 7 Fourier series Syllabus section...

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