ABSTRACT
This chapter discusses Fourier methods in the theory of differential equations. The Fourier coefficients of an integrable function is defined conveniently on a specific intervals or notation. From the point of view of pure mathematics, this complex notation has been shown to be useful, and it has become standardized. The chapter reviews the Dirichlet problem on the disc by studying the two-dimensional Laplace equation. This is probably the most important differential equation of mathematical physics. It describes a steady-state heat distribution, electrical fields, and many other important phenomena of nature. The differential equation is written in polar coordinates. A formal procedure with the Poisson integral series is presented for solving the Dirichlet problem.
