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Mathematics for the Physical Sciences

ISBN-10: 0486466620
ISBN-13: 9780486466620
Edition: 2008
Authors: Laurent Schwartz
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Description: In its discussions of mathematical entities and their elementary properties, this text employs examples from the physical sciences. Its concise treatment covers preliminary results in the integral calculus, distribution theory, Fourier series and  More...

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Book details

List price: $19.95
Copyright year: 2008
Publisher: Dover Publications, Incorporated
Publication date: 4/21/2008
Binding: Paperback
Pages: 368
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 1.078
Language: English

In its discussions of mathematical entities and their elementary properties, this text employs examples from the physical sciences. Its concise treatment covers preliminary results in the integral calculus, distribution theory, Fourier series and transforms, the Laplace transform, the wave and heat conduction equations, and gamma and Bessel functions. 1966 edition.

Preface
Preliminary results in the integral calculus: series and integrals
Preliminary Results on Series
Summable series
Semi-convergent series
Preliminary Results on Integration
The Lebesgue integral
Improper semi-convergent Lebesgue integrals
Functions Represented by Series and Integrals
Functions represented by series
Functions represented by integrals
Exercises for Chapter I
Elementary theory of distributions
Definition of Distributions
The vector space D
Distributions
The support of a distribution
Differentiation of Distributions
Definition
Examples of derivatives in the one-dimensional case
Examples of derivatives in the case of several variables
Multiplication of Distributions
Topology in Distribution Space. Convergence of Distributions. Series of Distributions
Distributions with Bounded Supports
Exercises for Chapter II
Convolution
Tensor Product of Distributions
Tensor product of two distributions
Tensor product of several distributions
Convolution
Convolution of two distributions
Definition of the convolution product of several distributions. Associativity of convolution
Convolution equations
Convolution in Physics
Exercises for Chapter III
Fourier series
Fourier Series of Periodic Functions and Distributions
Fourier series expansion of a periodic function
Fourier series expansion of a periodic distribution
Convergence of Fourier Series in the Distribution Sense and in the Function Sense
Convergence of the Fourier series of a distribution
Convergence of the Fourier series of a function
Hilbert Bases of a Hilbert Space. Mean-Square Convergence of a Fourier Series
Definition of a Hilbert space
Hilbert basis
The space L[superscript 2](T)
The Convolution Algebra D'([gamma])
Exercises for Chapter IV
The fourier transform
Fourier Transforms of Functions of One Variable
Introduction
Fourier transform
Fundamental relations and inequalities
Spaces s of infinitely differentiable functions with all derivatives decreasing rapidly
Fourier Transforms of Distributions in One Variable
Definition
Tempered distributions: the space y'
Fourier transforms of tempered distributions
The Parseval-Plancherel equation. Fourier transforms in L[superscript 2]
The Poisson summation formula
The Fourier transform: multiplication and convolution
Other expressions for the Fourier transform
Fourier Transforms in Several Variables
A Physical Application of the Fourier Transform: Solution of the Heat Conduction Equation
Exercises for Chapter V
The laplace transform
Laplace Transforms of Functions
Laplace Transforms of Distributions
Definition
Examples of Laplace transforms
The Laplace transforms and convolution
Fourier and Laplace transform. Inversion of the Laplace transform
Applications of the Laplace Transform. Operational Calculus
Exercises for Chapter VI
The wave and heat conduction equations
Equation of Vibrating Strings
Physical problems associated with the equation of vibrating strings
Solution of the equation of vibrating strings by the method of travelling waves. Cauchy's problem
Solution of Cauchy's problem by Fourier analysis
Vibrating Membranes and Waves in Three Dimensions
The solution of the vibrating membrane equation and the wave equation in three dimensions by the method of travelling waves. Cauchy problems
Solution of the Cauchy problem for vibrating membranes by the method of harmonics
Particular cases of rectangular and circular membranes
The wave equation in R[superscript n]
The Heat Conduction Equation
Solution by the method of propagation. Cauchy's problem
The solution of Cauchy's problem by the method of harmonics
Exercises for Chapter VII
The gamma function
The Function [Gamma] (z)
The Function B (p, q)
The Complementary Formula
Generalization of the Beta Function
Graphical Representation of the Function y = [Gamma](x) for Real x
Stirling's Formula
Application to the Expansion of 1/[Gamma] as an Infinite Product
The Function [psi](z) = [Gamma]'(z)/[Gamma](z)
Applications
Exercises for Chapter VIII
Bessel functions
Definitions and Elementary Properties
Definitions of the Bessel, Neumann and Hankel functions
Integral representations of Bessel functions
Recurrence relations
Other properties of Bessel functions
Formulae
Exercises for Chapter IX
Index

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