Fourier half-range expansions

In various applications, there is a practical need to use Fourier series in connection with function/(x) that are given on some intervals only. We could extend/(a ) periodically with period L and then represent the extended function by a Fourier series, which in general would involve both cosine and sine terms. 

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The detailed solution for Eg. flO.SI will not be shown here only the final result will be presented. The details will be left as an exercise for the reader. Keep in mind that the procedure for solving such a type of PDE is similar to that shown in Sec. 10.7. However, in the final stage, the expansion for f (x), as the initial condition, should become eguivalent to the half-range Fourier series, namely, the Fourier cosine series. 

Figure 7.15 shows a plot of the data compared with only two of the sin() Fourier components, which are the fundamental of the quarter eyele analysis and the third harmonic. The figure shows that these two sin() components give a good fit to the data. The use of sub-harmonics or half- and quarter-range Fourier expansions ean be a very useful data fitting technique for cases where one is interested only in a limited range of data and where such an extension of the data provides a much smoother function for Fourier analysis. 

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