Expansion Of Eigenfunctions With Separable Boundary Conditions

The topic of this paper is the properties of the eigenvalues and eigenfunctions ofSturm-Liouville problem on separable boundary conditions,we get the Green's functionsof which we make use to get the Fourier series of eigenfunctions on separable boundaryconditions.This article is divided into four parts.In the first part,the background of the paper,some methods and results,are intro-duced brie?y.In next part,some important concepts and theorems which are related to the articleare given.Since they are well-known to everyone,the article only gives the results butnot prove them.The third part contains some lemmas and parts of their proofs which are necessaryto the proof of the main theorem.The last part gives the main theorem and its proof in the article,at the same timesome corollaries are obtained easily.