Transcription of Introduction to Sturm-Liouville Theory
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Introduction to Sturm-Liouville Theory
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, ..) are pairwiseorthogonal on [0, ] relative to the weight functionw(x) the bessel function of the first kind of orderm, andlet mndenote itsnth positive zero.
orthogonal on [0,π] relative to the weight function w(x) ≡ 1. 2 Let J m be the Bessel function of the first kind of order m, and let α mn denote its nth positive zero. Then the functions f n(x) = J m(α mnx/a) are pairwise orthogonal on [0,a] with respect to the weight function w(x) = x. 3 The functions f 0(x) = 1, f 1(x) = 2x, f 2(x ...
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