# Math 4510

 The flashcards below were created by user 85shelbyc on FreezingBlue Flashcards. Properties of the Regular Sturm-Liouville Problem 1) There exists an infinite number of eigenvalues that can be arranged in increasing order such that as . 2) For each eigenvalue there is only one eigenfunction (except for non-zero constant multiples) 3) Eigenfunctions corresponding to different eigenvalues are linearly independent 4) The set of eigenfunctions corresponding to the set of eigenvalues is orthongonal with respect to the weight function p(x) on the interval [a,b]. Regular Sterm-Liouville Problem where r, r', p, and q are continuous on the interval (a,b) and p(x)>0, r(x)>0 for all x in the interval (a,b) Subject to: A1y(a) + B1y'(a) = 0, A2y(b) + B2y'(b) = 0A12 + B12 > 0, A22 + B22 > 0 Convergence of a Fourier Series Let f and f ' be piecewise continuous on some interval [-p,p]. That is, let f and f ' be continuous except at a finite number of points on the interval and have only finite discontinuities at these points. Then the Fourier series of f on the interval converges to f(x) at a point of continuity. At a point of discontinuity the Fourier series converges to the average Fourier Series The Fourier series of a function f defined by the interval (-p,p) is given by: where:   Orthogonal Functions Definition Two functions f(x) and g(x) are orthogonal on an interval [a,b] if Existence of Power Series Solutions Theorem If x = x0 is an ordinary point of the differential equation , we can always find two linearly independent solutions in the form of a power series, centered at x0. A series solution converges on some interval defined by |x-x0| < R, where R is the distance from x0 to the closest singular point. Frobenius' Theorem If x = x0 is a regular singular point of the differential equation, then there exists at least one solution of the form where the number r is constant to be determined. The series will converge on some interval 0 < x-x0 < R Definition of Gamma Function , x > 0 Bessel Equation Bessel Functions of the First Kind J-v = ... Author85shelbyc ID138586 Card SetMath 4510 DescriptionTheorems and Definitions Updated2012-02-29T13:22:28Z Show Answers