Orthogonal Functions

It was the purpose of this thesis (1) to investigate certain known orthogonal functions; (2) to exhibit elements of similarity among them; (3) to show that they arise as the solutions of differential equations in a similar way; (4) to establish that the differential equations which yield them are similar and are special cases of yet another more general differential equation; (5) to verify that the Sturm-Liouville theory can be applied to each of these differential equations; and (6) to demonstrate how each of these orthogonal functions is used to give a series representation of a given function. By direct computation it was shown that the orthogonal functions studied arose from their respective differential equations in a similar way. These differential equations were then shown to be special cases of a more general differential equation, thus showing another similarity among the orthogonal functions. The Sturm-Liouville theory was used to find certain information in order to establish the orthogonality of the functions that were a solution of the various differential equations. Finally, by direct methods each of the orthogonal functions was utilized in forming a series representation of two arbitrary functions and these series representations were put in graphic form to show yet another similarity of the orthogonal functions that were studied.