An Introduction to Fourier Series and Integrals

Robert T. Seeley

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486151793
• Topic: Mathematics
• Subtopic: Mathematics Reference
• Index: Mathematics

A compact, sophomore-to-senior-level guide, Dr. Seeley's text introduces Fourier series in the way that Joseph Fourier himself used them: as solutions of the heat equation in a disk. Emphasizing the relationship between physics and mathematics, Dr. Seeley focuses on results of greatest significance to modern readers.
Starting with a physical problem, Dr. Seeley sets up and analyzes the mathematical modes, establishes the principal properties, and then proceeds to apply these results and methods to new situations. The chapter on Fourier transforms derives analogs of the results obtained for Fourier series, which the author applies to the analysis of a problem of heat conduction. Numerous computational and theoretical problems appear throughout the text.

1

Dirichlet's Problem and
Poisson’s Theorem

This entire chapter discusses a single problem that lies at the heart of the subject of Fourier series. In physical terms, it is to determine the steady state temperature distribution in a disk when the boundary temperatures are known; the mathematical formulation is known as Dirichlet's problem. Starting from scratch, we arrive at a solution in the form of a series Σ a_nr^|n|e^inθ, called a trigonometric series. The attempt to verify that this can actually be made to solve the heat problem leads to Poisson’s theorem, one of the most important and attractive elementary theorems of analysis. A number of important consequences are evolved below, but these few only suggest the importance of Poisson’s theorem. Echoes of the proof, and of the integral representation on which it is based, appear frequently in mathematics even today.

1-1. THE EQUATION OF STEADY STATE HEAT CONDUCTION

The first step in solving heat problems (as in most problems of mathematical physics) is to find a differential equation governing the situation. Since we are concerned with a disk, the natural coordinates are polar. The temperature at the point with coordinates (r, θ) is denoted by u(r, θ).

In order to find the relevant equation, consider any section of the disk given by

(see Fig. 1-1). Since we are considering a steady state, the rate at which heat flows into this section must be 0; otherwise the average temperature would change with time. Now it is a basic postulate of heat conduction that the rate at which heat crosses a curve C is proportional to the integral along C of the normal derivative ∂u/∂n of the temperature distribution. Here ∂u/∂n is the derivative of u with respect to arc length along any curve perpendicular to C. When C is the side θ — θ₁ of the portion given in (1-1), we can take these perpendicular curves to be given by r = constant. Then, since the length of a circular arc is the angle times the radius, along

FIGURE 1-1.Derivation of the heat equation in polar coordinates.

θ = θ₁ the normal derivative is

and the rate at which heat flows into the section in Fig. 1-1 along the boundary θ = θ₁ is

where k is the conductivity. Adding corresponding expressions for the other three boundaries and setting the net flow equal to zero, we get

If we divide by θ₁ — θ₀ and let θ₁ θ₀this yields

(The first term comes from Leibnitz’s rule, and the second from the fundam...