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6.1 A Theory of Eigenfunction Expansions

We have solved many initial-boundary value problems in which separation of variables has led to sines and/or cosines as eigenfunctions, in terms of which we had to write some given function (such as an initial temperature, position or velocity) in a Fourier expansion.

We will now solve initial-boundary value problems in which “new” functions, or special functions, have to be created as eigenfunctions. Further, we will have to expand some given function in a series of these new eigenfunctions. This section provides an environment in which we can deal with these issues.

Begin with a general form of the differential equation for the eigenfunctions. Separation of variables in many kinds of important problems leads to a differential equation of the form

(6.1) where r, q, and p are given, continuous functions on the relevant interval, say [a, b] or (a, b), and r(x) > 0 and p(x) > 0 for a < x < b.

This is the Sturm-Liouville differential equation. Of course, any differential equation can be written in different ways, so we will refer to the form given in equation 6.1 as the standard form of the Sturm-Liouville equation.

A Sturm-Liouville problem consists of this differential equation, together with boundary conditions. We will consider two kinds of boundary conditions.

A regular Sturm-Liouville ...

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