, can be computed by solving the equation tan(πλ)=-λ. Graphically, the solutions are illustrated in the following plot.
Eigenfunction Expansions
William O. Bray
The theme of this notebook is to illustrate the behavior of eigenfunction expansions. We consider the Sturm-Liouville problem
ODE: X''+μX=0, 0<x<π
BC: X(0)=0, X’(π)+X(π)=0
The eigenvalues are all positive and writing , can be computed by solving the equation tan(πλ)=-λ. Graphically, the solutions are illustrated in the following plot.
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It is clear that 
. The first thirty values of λ are estimated as follows.
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The eigenfunctions are given by 
. A plot of a few of these is given as follows.
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NOTE: While each eigenfunction is a periodic function (on the real line), the collection of such does not have a common period as in the case of classical Fourier series.
Given a function 
, the Fourier series of interest has the form:
, 
, 
For later use the following commands are useful.
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Example 1
Consider the piecewise continuous function
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We can compute the Fourier coefficients the following way.
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Example 2
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Example 3
The following example illustrates the uniform convergence under the conditions that the function is 
and satisfies the boundary conditions.
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