# 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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(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.