Transcription of ORTHOGONAL FUNCTIONS: THE LEGENDRE, - LSUMath
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ORTHOGONAL FUNCTIONS: THE LEGENDRE, - LSUMath
ORTHOGONAL functions : THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALSTHOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR,AND TYLER legendre , Laguerre, and Hermite equations areall homogeneous second order sturm -Liouville equations. Usingthe sturm -Liouville Theory we will be able to show that polynomialsolutions to these equations are ORTHOGONAL . In a more generalcontext, finding that these solutions are ORTHOGONAL allows us towrite a function as a Fourier series with respect to these legendre , Laguerre, and Hermite equations have many realworld practical uses which we will not discuss here. We will only focuson the methods of solution and use in a mathematical sense. In solvingthese equations explicit solutions cannot be found. That is solutionsin in terms of elementary functions cannot be found. In many cases itis easier to find a numerical or series is a generalized Fourier series theory which allows one to writea functionf(x) as a linear combination of an ORTHOGONAL system offunctions 1(x), 2(x).
THOMAS2 COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO 2. The Sturm-Liouville Theory A Sturm-Liouville equation is a homogeneous second order di eren-
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