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Preface | |

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Preliminary results in the integral calculus: series and integrals | |

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Preliminary Results on Series | |

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Summable series | |

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Semi-convergent series | |

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Preliminary Results on Integration | |

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The Lebesgue integral | |

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Improper semi-convergent Lebesgue integrals | |

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Functions Represented by Series and Integrals | |

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Functions represented by series | |

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Functions represented by integrals | |

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Exercises for Chapter I | |

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Elementary theory of distributions | |

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Definition of Distributions | |

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The vector space D | |

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Distributions | |

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The support of a distribution | |

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Differentiation of Distributions | |

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Definition | |

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Examples of derivatives in the one-dimensional case | |

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Examples of derivatives in the case of several variables | |

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Multiplication of Distributions | |

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Topology in Distribution Space. Convergence of Distributions. Series of Distributions | |

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Distributions with Bounded Supports | |

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Exercises for Chapter II | |

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Convolution | |

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Tensor Product of Distributions | |

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Tensor product of two distributions | |

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Tensor product of several distributions | |

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Convolution | |

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Convolution of two distributions | |

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Definition of the convolution product of several distributions. Associativity of convolution | |

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Convolution equations | |

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Convolution in Physics | |

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Exercises for Chapter III | |

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Fourier series | |

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Fourier Series of Periodic Functions and Distributions | |

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Fourier series expansion of a periodic function | |

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Fourier series expansion of a periodic distribution | |

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Convergence of Fourier Series in the Distribution Sense and in the Function Sense | |

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Convergence of the Fourier series of a distribution | |

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Convergence of the Fourier series of a function | |

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Hilbert Bases of a Hilbert Space. Mean-Square Convergence of a Fourier Series | |

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Definition of a Hilbert space | |

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Hilbert basis | |

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The space L[superscript 2](T) | |

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The Convolution Algebra D'([gamma]) | |

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Exercises for Chapter IV | |

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The fourier transform | |

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Fourier Transforms of Functions of One Variable | |

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Introduction | |

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Fourier transform | |

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Fundamental relations and inequalities | |

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Spaces s of infinitely differentiable functions with all derivatives decreasing rapidly | |

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Fourier Transforms of Distributions in One Variable | |

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Definition | |

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Tempered distributions: the space y' | |

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Fourier transforms of tempered distributions | |

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The Parseval-Plancherel equation. Fourier transforms in L[superscript 2] | |

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The Poisson summation formula | |

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The Fourier transform: multiplication and convolution | |

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Other expressions for the Fourier transform | |

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Fourier Transforms in Several Variables | |

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A Physical Application of the Fourier Transform: Solution of the Heat Conduction Equation | |

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Exercises for Chapter V | |

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The laplace transform | |

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Laplace Transforms of Functions | |

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Laplace Transforms of Distributions | |

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Definition | |

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Examples of Laplace transforms | |

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The Laplace transforms and convolution | |

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Fourier and Laplace transform. Inversion of the Laplace transform | |

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Applications of the Laplace Transform. Operational Calculus | |

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Exercises for Chapter VI | |

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The wave and heat conduction equations | |

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Equation of Vibrating Strings | |

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Physical problems associated with the equation of vibrating strings | |

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Solution of the equation of vibrating strings by the method of travelling waves. Cauchy's problem | |

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Solution of Cauchy's problem by Fourier analysis | |

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Vibrating Membranes and Waves in Three Dimensions | |

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The solution of the vibrating membrane equation and the wave equation in three dimensions by the method of travelling waves. Cauchy problems | |

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Solution of the Cauchy problem for vibrating membranes by the method of harmonics | |

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Particular cases of rectangular and circular membranes | |

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The wave equation in R[superscript n] | |

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The Heat Conduction Equation | |

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Solution by the method of propagation. Cauchy's problem | |

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The solution of Cauchy's problem by the method of harmonics | |

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Exercises for Chapter VII | |

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The gamma function | |

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The Function [Gamma] (z) | |

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The Function B (p, q) | |

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The Complementary Formula | |

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Generalization of the Beta Function | |

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Graphical Representation of the Function y = [Gamma](x) for Real x | |

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Stirling's Formula | |

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Application to the Expansion of 1/[Gamma] as an Infinite Product | |

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The Function [psi](z) = [Gamma]'(z)/[Gamma](z) | |

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Applications | |

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Exercises for Chapter VIII | |

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Bessel functions | |

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Definitions and Elementary Properties | |

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Definitions of the Bessel, Neumann and Hankel functions | |

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Integral representations of Bessel functions | |

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Recurrence relations | |

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Other properties of Bessel functions | |

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Formulae | |

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Exercises for Chapter IX | |

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Index | |