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    Fourier series of sqrt(R^2 - x^2)

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    Please determine the Fourier series for the attached function.

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    Solution Preview

    To compute the Fourier series, let's start by defining our set of orthonormal basis functions. We need to have a complete set of basis functions with a period of 2 R. We can take these to be the functions e_n(x) for integer n defined as:

    e_n(x) = exp(n pi i/R x)

    If we define the inner product of two functions f and g as:

    <f,g> = 1/(2R) Integral from -R to R of f(x) g*(x) dx

    then <e_n,e_m> = delta_{n,m}, therefore the e_n(x) form an orthonormal set of basis functions. If f(x) is a function with a period of 2 R, we have:

    f(x) = Sum over n from minus infinity to infinity of <f,e_n> e_n(x) (1)

    If f(x) is an even function, then we have:

    <f,e_n> = 1/(2R) Integral from -R to R of f(x)e_n*(x)dx = 1/(2R) Integral from -R to R of f(x) exp(-n ...

    Solution Summary

    A detailed calculation of the Fourier series is given.