Mathematics for the Physical Sciences

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ISBN: 9780486466620 | 0486466620
Cover: Paperback
Copyright: 4/21/2008

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This exploration of the mathematical methods of physics takes a careful look at mathematical entities and explains their elementary properties. Its examples, drawn from the physical sciences, illustrate the application of concepts. The theory of distributions is introduced early and employed throughout the text.

Preface	p. 11
Preliminary results in the integral calculus: series and integrals
Preliminary Results on Series	p. 13
Summable series	p. 13
Semi-convergent series	p. 23
Preliminary Results on Integration	p. 26
The Lebesgue integral	p. 26
Improper semi-convergent Lebesgue integrals	p. 43
Functions Represented by Series and Integrals	p. 46
Functions represented by series	p. 46
Functions represented by integrals	p. 57
Exercises for Chapter I	p. 65
Elementary theory of distributions
Definition of Distributions	p. 71
The vector space D	p. 71
Distributions	p. 73
The support of a distribution	p. 79
Differentiation of Distributions	p. 80
Definition	p. 80
Examples of derivatives in the one-dimensional case	p. 82
Examples of derivatives in the case of several variables	p. 85
Multiplication of Distributions	p. 91
Topology in Distribution Space. Convergence of Distributions. Series of Distributions	p. 94
Distributions with Bounded Supports	p. 98
Exercises for Chapter II	p. 100
Convolution
Tensor Product of Distributions	p. 110
Tensor product of two distributions	p. 110
Tensor product of several distributions	p. 112
Convolution	p. 112
Convolution of two distributions	p. 112
Definition of the convolution product of several distributions. Associativity of convolution	p. 121
Convolution equations	p. 123
Convolution in Physics	p. 134
Exercises for Chapter III	p. 140
Fourier series
Fourier Series of Periodic Functions and Distributions	p. 145
Fourier series expansion of a periodic function	p. 145
Fourier series expansion of a periodic distribution	p. 149
Convergence of Fourier Series in the Distribution Sense and in the Function Sense	p. 152
Convergence of the Fourier series of a distribution	p. 152
Convergence of the Fourier series of a function	p. 154
Hilbert Bases of a Hilbert Space. Mean-Square Convergence of a Fourier Series	p. 158
Definition of a Hilbert space	p. 158
Hilbert basis	p. 159
The space L[superscript 2](T)	p. 160
The Convolution Algebra D'([gamma])	p. 164
Exercises for Chapter IV	p. 170
The fourier transform
Fourier Transforms of Functions of One Variable	p. 178
Introduction	p. 178
Fourier transform	p. 179
Fundamental relations and inequalities	p. 180
Spaces s of infinitely differentiable functions with all derivatives decreasing rapidly	p. 182
Fourier Transforms of Distributions in One Variable	p. 187
Definition	p. 187
Tempered distributions: the space y'	p. 188
Fourier transforms of tempered distributions	p. 189
The Parseval-Plancherel equation. Fourier transforms in L[superscript 2]	p. 194
The Poisson summation formula	p. 196
The Fourier transform: multiplication and convolution	p. 197
Other expressions for the Fourier transform	p. 199
Fourier Transforms in Several Variables	p. 200
A Physical Application of the Fourier Transform: Solution of the Heat Conduction Equation	p. 205
Exercises for Chapter V	p. 208
The laplace transform
Laplace Transforms of Functions	p. 215
Laplace Transforms of Distributions	p. 217
Definition	p. 217
Examples of Laplace transforms	p. 218
The Laplace transforms and convolution	p. 222
Fourier and Laplace transform. Inversion of the Laplace transform	p. 224
Applications of the Laplace Transform. Operational Calculus	p. 230
Exercises for Chapter VI	p. 235
The wave and heat conduction equations
Equation of Vibrating Strings	p. 242
Physical problems associated with the equation of vibrating strings	p. 242
Solution of the equation of vibrating strings by the method of travelling waves. Cauchy's problem	p. 251
Solution of Cauchy's problem by Fourier analysis	p. 270
Vibrating Membranes and Waves in Three Dimensions	p. 280
The solution of the vibrating membrane equation and the wave equation in three dimensions by the method of travelling waves. Cauchy problems	p. 281
Solution of the Cauchy problem for vibrating membranes by the method of harmonics	p. 291
Particular cases of rectangular and circular membranes	p. 293
The wave equation in R[superscript n]	p. 298
The Heat Conduction Equation	p. 298
Solution by the method of propagation. Cauchy's problem	p. 298
The solution of Cauchy's problem by the method of harmonics	p. 301
Exercises for Chapter VII	p. 303
The gamma function
The Function [Gamma] (z)	p. 311
The Function B (p, q)	p. 313
The Complementary Formula	p. 315
Generalization of the Beta Function	p. 317
Graphical Representation of the Function y = [Gamma](x) for Real x	p. 318
Stirling's Formula	p. 320
Application to the Expansion of 1/[Gamma] as an Infinite Product	p. 322
The Function [psi](z) = [Gamma]'(z)/[Gamma](z)	p. 326
Applications	p. 327
Exercises for Chapter VIII	p. 330
Bessel functions
Definitions and Elementary Properties	p. 334
Definitions of the Bessel, Neumann and Hankel functions	p. 334
Integral representations of Bessel functions	p. 341
Recurrence relations	p. 343
Other properties of Bessel functions	p. 345
Formulae	p. 350
Exercises for Chapter IX	p. 353
Index	p. 357
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Mathematics for the Physical Sciences

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