Fourier series and orthogonal polynomials / by Dunham Jackson.

Format:

Book

Author/Creator:

Jackson, Dunham, 1888-1946, author.

Series:

Carus mathematical monographs ; Number 6.

Carus Mathematical Monographs ; Number 6

Language:

English

Subjects (All):

Fourier series.

Orthogonal polynomials.

Fourier analysis.

Physical Description:

1 online resource (xii, 234 pages) : digital, PDF file(s).

Edition:

1st ed.

Other Title:

Fourier Series & Orthogonal Polynomials

Orthogonal polynomials.

Place of Publication:

Washington : Mathematical Association of America, 1941.

Language Note:

English

Summary:

The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. The underlying theme of this monograph is that the fundamental simplicity of the properties of orthogonal functions and the developments in series associated with them makes those functions important areas of study for students of both pure and applied mathematics. The book starts with Fourier series and goes on to Legendre polynomials and Bessel functions. Jackson considers a variety of boundary value problems using Fourier series and Laplace's equation. Chapter VI is an overview of Pearson frequency functions. Chapters on orthogonal, Jacobi, Hermite, and Laguerre functions follow. The final chapter deals with convergence. There is a set of exercises and a bibliography. For the reading of most of the book no specific preparation is required beyond a first course in the calculus. A certain amount of 'mathematical maturity' is presupposed, or should be acquired in the course of the reading.

Contents:

""Front Cover""; ""Fourier Series and Orthogonal Polynomials""; ""Copyright Page""; ""Preface ""; ""Table of Contents""; ""Chapter I. Fourier Series""; ""1. Definition of Fourier series""; ""2. Orthogonality of sines and cosines""; ""3. Determination of the coefficients""; ""4. Series of cosines and series of sines""; ""5. Examples""; ""6. Magnitude of coefficients under special hypotheses""; ""7. Riemann's theorem on limit of general coefficient""; ""8. Evaluation of a sum of cosines""; ""9. Integral formula for partial sum of Fourier series""; ""10. Convergence at a point of continuity""

""1. Preliminary orientation""""2. Definition of the Legendre polynomials by means of the generating function""; ""3. Recurrence formula""; ""4. Differential equation and related formulas""; ""5. Orthogonality""; ""6. Normalizing factor""; ""7. Expansion of an arbitrary function in series""; ""8. Christoffel's identity""; ""9. Solution of the differential equation""; ""10. Rodrigues's formula""; ""11. Integral representation""; ""12. Bounds of Pn(x)""; ""13. Convergence at a point of continuity interior to the interval""

""14. Convergence at a point of discontinuity interior to the interval""""Chapter III. Bessel Functions""; ""1. Preliminary orientation""; ""2. Definition of J0(x)""; ""3. Orthogonality""; ""4. Integral representation of Jo{x)""; ""5. Zeros of J0(x) and related functions""; ""6. Expansion of an arbitrary function in series""; ""7. Definition of Jn(x)""; ""8. Orthogonality: developments in series""; ""9. Integral representation of Jn(x)""; ""10. Recurrence formulas""; ""11. Zeros""; ""12. Asymptotic formula""; ""13. Orthogonal functions arising from linear boundary value problems""

""Chapter IV. Boundary Value Problems""""1. Fourier series: Laplace's equation in an infinite strip""; ""2. Fourier series: Laplace's equation in a rectangle""; ""3. Fourier series: vibrating string""; ""4. Fourier series: damped vibrating string""; ""5. Polar coordinates in the plane""; ""6. Fourier series: Laplace's equation in a circle; Poisson's integral""; ""7. Transformation of Laplace's equation in three dimensions""; ""8. Legendre series: Laplace's equation in a sphere""; ""9. Bessel series: Laplace's equation in a cylinder""; ""10. Bessel series: circular drumhead""

""Chapter V. Double Series

Notes:

Title from publisher's bibliographic system (viewed on 02 Oct 2015).

Includes bibliographical references and index.

Description based on print version record.

ISBN:

0-88385-006-0

1-61444-006-9