An Introduction to Fourier Series and Integrals
eBook - ePub

An Introduction to Fourier Series and Integrals

Robert T. Seeley

Partager le livre
  1. 112 pages
  2. English
  3. ePUB (adapté aux mobiles)
  4. Disponible sur iOS et Android
eBook - ePub

An Introduction to Fourier Series and Integrals

Robert T. Seeley

DĂ©tails du livre
Aperçu du livre
Table des matiĂšres
Citations

À propos de ce livre

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.

Foire aux questions

Comment puis-je résilier mon abonnement ?
Il vous suffit de vous rendre dans la section compte dans paramĂštres et de cliquer sur « RĂ©silier l’abonnement ». C’est aussi simple que cela ! Une fois que vous aurez rĂ©siliĂ© votre abonnement, il restera actif pour le reste de la pĂ©riode pour laquelle vous avez payĂ©. DĂ©couvrez-en plus ici.
Puis-je / comment puis-je télécharger des livres ?
Pour le moment, tous nos livres en format ePub adaptĂ©s aux mobiles peuvent ĂȘtre tĂ©lĂ©chargĂ©s via l’application. La plupart de nos PDF sont Ă©galement disponibles en tĂ©lĂ©chargement et les autres seront tĂ©lĂ©chargeables trĂšs prochainement. DĂ©couvrez-en plus ici.
Quelle est la différence entre les formules tarifaires ?
Les deux abonnements vous donnent un accĂšs complet Ă  la bibliothĂšque et Ă  toutes les fonctionnalitĂ©s de Perlego. Les seules diffĂ©rences sont les tarifs ainsi que la pĂ©riode d’abonnement : avec l’abonnement annuel, vous Ă©conomiserez environ 30 % par rapport Ă  12 mois d’abonnement mensuel.
Qu’est-ce que Perlego ?
Nous sommes un service d’abonnement Ă  des ouvrages universitaires en ligne, oĂč vous pouvez accĂ©der Ă  toute une bibliothĂšque pour un prix infĂ©rieur Ă  celui d’un seul livre par mois. Avec plus d’un million de livres sur plus de 1 000 sujets, nous avons ce qu’il vous faut ! DĂ©couvrez-en plus ici.
Prenez-vous en charge la synthÚse vocale ?
Recherchez le symbole Écouter sur votre prochain livre pour voir si vous pouvez l’écouter. L’outil Écouter lit le texte Ă  haute voix pour vous, en surlignant le passage qui est en cours de lecture. Vous pouvez le mettre sur pause, l’accĂ©lĂ©rer ou le ralentir. DĂ©couvrez-en plus ici.
Est-ce que An Introduction to Fourier Series and Integrals est un PDF/ePUB en ligne ?
Oui, vous pouvez accĂ©der Ă  An Introduction to Fourier Series and Integrals par Robert T. Seeley en format PDF et/ou ePUB ainsi qu’à d’autres livres populaires dans Mathematics et Mathematics Reference. Nous disposons de plus d’un million d’ouvrages Ă  dĂ©couvrir dans notre catalogue.

Informations

Année
2014
ISBN
9780486151793
1
Image
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 anr|n|einξ, 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, Ξ) is denoted by u(r, Ξ).
In order to find the relevant equation, consider any section of the disk given by
image
(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 ξ — ξ1 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
image
FIGURE 1-1.Derivation of the heat equation in polar coordinates.
Ξ = Ξ1 the normal derivative is
image
and the rate at which heat flows into the section in Fig. 1-1 along the boundary Ξ = Ξ1 is
image
where k is the conductivity. Adding corresponding expressions for the other three boundaries and setting the net flow equal to zero, we get
image
If we divide by ξ1 — ξ0 and let ξ1 ξ0this yields
image
(The first term comes from Leibnitz’s rule, and the second from the fundam...

Table des matiĂšres