Transcription of Introduction to Sturm-Liouville Theory - Trinity University
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Introduction to Sturm-Liouville Theory - Trinity University
OrthogonalitySturm- liouville problemsEigenvalues and eigenfunctionsIntroduction to Sturm-Liouville TheoryRyan C. DailedaTrinity UniversityPartial Differential EquationsApril 10, 2012 DailedaSturm- liouville TheoryOrthogonalitySturm- liouville problemsEigenvalues and eigenfunctionsInner products with weight functionsSuppose thatw(x) is a nonnegative function on [a,b]. Iff(x) andg(x) are real-valued functions on [a,b] we define theirinnerproduct on[a,b]with respect to the weightwto behf,gi= baf(x)g(x)w(x) sayfandgareorthogonal on[a,b]with respect to theweightwifhf,gi= :The inner product and orthogonality depend on the choice ofa, (x) 1, these definitions reduce to the ordinary TheoryOrthogonalitySturm- liouville problemsEigenvalues and eigenfunctionsExamples1 The functionsfn(x) = sin(nx) (n= 1,2.)
Orthogonality Sturm-Liouville problems Eigenvalues and eigenfunctions Inner products with weight functions Suppose that w(x) is a nonnegative function on [a,b].
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