Advanced Engineering Mathematics, Lecture 4.6: Some special orthogonal functions

Advanced Engineering Mathematics, Lecture 4.6: Some special orthogonal functions.

In this lecture, we revisit some ODEs from physics and engineering that we saw when learning how to find generalized power series solutions by the Frobenius method. We study four of them in detail: Legendre's equation, parametric Bessel's equation, Chebyshev's equation, and Hermite's equation. All of these are secretly Sturm-Liouville problems, and in this lecture, we learn (without actually solving them) what their eigenvalues and eigenfunctions are. In each case, we looks at the resulting generalized Fourier series. In three of these cases, this means that we can express arbitrary functions (with mild conditions) as an infinite sum of eigenfunctions, which happen to be polynomials. Though you likely have have heard of these before -- Legendre, Chebyshev, and Hermite polynomials, you may not have realized how they arise as eigenfunctions of Sturm-Liouville problems from physical models.

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[الأستاذبوكليةأحمدأيوب]

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