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    Sturm-Liouville expansion

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    3. Expand... in terms of the eigenfunctions of the Sturm-Liouville problem.

    Please see attached.

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    https://brainmass.com/math/fourier-analysis/sturm-liouville-expansion-44174

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    (The first 3 non-vanishing terms, not the first four non-vanishing terms)

    (H(x) denotes Heaviside Step Function. We can take any Heaviside Step Function.)

    (The problem is from Periodic and Singular Sturm-Liouville Problems.)

    This is the famous Sturm-Liouville equation that leads to the Legendre's polynomials.

    The equation can be written as:

    The general orthogonality of the eigenfunctions:

    Assume the general form of Sturm-Liouville equation:

    Define two eigenfunctions which ...

    Solution Summary

    This shows how to expand a function in terms of the eigenfunctions of a given Sturm-Liouville problem and plot the function and the sum of the first three nonvanishing terms.

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