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The Volterra functions appeared at the beginning of the second decade of the twentieth century in the theory of definite integrals, integral equations and prime numbers in the works of famous mathematicians Srinivasa Ramanujan, Jacques Touchard, Vito Volterra and Edmund Landau. However, between 1943-1953, the Volterra functions started to play an important role also in the investigations of a number of French mathematicians because they found that these functions are the direct and inverse transforms in the laplace transformation of some elementary and special functions. This book examines both integral equations and Volterra functions in detail and highlights the important roles that each of these functions perform. ... Read more

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ntegrals, and illustrates the application if this method to electromagnetics problems. Once the basics have been mastered, one quickly realizes that the method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the present method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. Two functions, the generalized hypergeometric function and the Meijer G-function, are very much related to the Mellin-transform method and arise frequently when the method is applied. Because these functions can be automatically handled by modern numerical routines, they are now much more useful than they were in the past.The Mellin-transform method and the two aforementioned functions are discussed first. Then the method is applied in three examples to obtain results, which, at least in the antenna/electromagnetics literature, are believed to be new. In the first example, a closed-form expression, as a generalized hypergeometric function, is obtained for the power radiated by a constant-current circular-loop antenna. The second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived. Additional examples are also briefly discussed. ... Read more

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This book introduces a new approach to the rapid numericalapplication of certain types of linear operators to arbitrary vectors.Thescheme is applicable to many classical integral transforms, and to theexpansions associated with most families of classical special functions,including, among others, Bessel functions, Legendre, Hermite, andLaguerre polynomials, Spherical Harmonics, and Prolate SpheroidalWave functions.Calculations of this type are encountered ingeophysical modeling, submarine warfare, quantum mechanicalcalculations, and simulation of radar phenomena. The performance ofthe algorithm is illustrated via a number of numerical examples.Thematerial presented in this book assumes a working knowledge of linearalgebra and calculus.Graduate level students and practitioners in thefields of scientific computing, numerical linear algebra, andmathematical physics will find this book useful. ... Read more

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This digital document is a journal article from International Journal of Production Economics, published by Elsevier in 2004. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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We sketch the integral transform method in a close connection with its historical development and survey newer applications to problems in production economics and finance. ... Read more

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In 1903 Fredholm published his famous paper on integral equations. Since then linear integral operators have become an important tool in many areas, including the theory of Fourier series and Fourier integrals, approximation theory and summability theory, and the theory of integral and differential equations. As regards the latter, applications were soon extended beyond linear operators. In approximation theory, however, applications were limited to linear operators mainly by the fact that the notion of singularity of an integral operator was closely connected with its linearity. This volume presents a comprehensive treatment of approximation theory by means of nonlinear integral operator in function spaces. In particular, the fundamental notions of approximate identity for kernels of nonlinear operators and a general concept of modulus of continuity are developed in order to obtain consistent approximation results. Applications to nonlinear summability, nonlinear integral equations and nonlinear sampling theory are given. In particular, the study of nonlinear sampling operators is important since the results permit the reconstruction of several classes of signals. ... Read more

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This book is based on two closely-related courses. The first of these courses is Integration and Metric Spaces, and the second being Functional Analysis. Though the contents of Functional Analysis have been used for both an undergraduate course and an introductory graduate course, this text is designed primarily for undergraduate students. The prerequisites of this book are deliberately modest, and it is assumed that the students have some familiarity with Introductory Calculus and Linear Algebra plus the basic (direct, indirect) proof methods. ... Read more

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The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures. ... Read more

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The major achievements of mathematical analysis from Newton and Euler to modern applications of mathematics in physical sciences, engineering and other areas are presented in this volume. Its three parts cover the methods of analysis: representation methods, asymptotic methods and transform methods. The authors - the well-known analysts M.A. Evgrafov and M.V. Fedoryuk - have not simply presented a compendium of techniques but have stressed throughout the underlying unity of the various methods. The fundamental ideas are clearly presented and illustrated with interesting and non-trivial examples. References, together with guides to the literature, are provided for those readers who wish to go further. ... Read more

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A self-contained presentation of the major areas of complex analysis that are referred to and used in applied mathematics and mathematical physics. Topics discussed include infinite products, ordinary differential equations and asymptotic methods. ... Read more

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This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics.

This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods. Familiarity with analysis (distributions and Fourier transformation) and differential geometry is useful. Additionally, this book is designed for a one-semester introductory course on Fourier integral operators aimed at a broad audience.

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This volume presents a state-of-the-art account of the theoryand applications of integral equations of convolution type, and ofcertain classes of integro-differential and non-linear integralequations. An extensive and well-motivated discussion of some openquestions and of various important directions for further research isalso presented. The book has been written so as to be self-contained,and includes a list of symbols with their definitions.
For users of convolution integral equations, the volume containsnumerous, well-classified inversion tables which correspond to thevarious convolutions and intervals of integration. It also has anextensive, up-to-date bibliography. The convolution integral equationswhich are considered arise naturally from a large variety of physicalsituations and it is felt that the types of solutions discussed willbe usefull in many diverse disciplines of applied mathematics andmathematical physical.
For researchers and graduate students in the mathematical and physicalsciences whose work involves the solution of integral equations.
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