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Books in Functions of a complex variable

  • Complex Variables

    • 1st Edition
    • H. R. Chillingworth
    • C. Plumpton
    • English
    Complex Variables focuses on the principles, characteristics, and functions of complex variables, as well as infinite series, complex numbers, and convergence and divergence. The book first examines complex numbers and the sequences and limits of point sets in the complex plane. Discussions focus on non-decreasing real sequences, boundedness of convergent sequences, boundary points, closed sets, bounded and unbounded sets in the complex plane, complex conjugates, complex numbers as an extension of the real number field, scalar multiplication, modulus, and number pairs. The manuscript then takes a look at the tests for convergence of infinite series, functions of a complex variable, and elementary functions. Concerns cover repeated differentiation of an infinite series, differentiability of power series, hyperbolic functions, link between the exponential and trigonometric functions, orthogonal families of curves, differentiability, testing for convergence or divergence, and series with negative or complex terms. The text examines miscellaneous theorems, contour integration, zeros and singularities, and integration, including order of magnitude of a function, infinite integrals involving trigonometric functions, and sum-limit and anti-differentiation... The publication is highly recommended for students and teachers wanting to explore complex variables.

  • Nine Introductions in Complex Analysis - Revised Edition

    • 1st Edition
    • Volume 208
    • Sanford L. Segal
    • English
    The book addresses many topics not usually in "second course in complex analysis" texts. It also contains multiple proofs of several central results, and it has a minor historical perspective.

  • Complex Numbers

    Lattice Simulation and Zeta Function Applications
    • 1st Edition
    • S C Roy
    • English
    An informative and useful account of complex numbers that includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the ever-elusory Riemann hypothesis. Stephen Roy assumes no detailed mathematical knowledge on the part of the reader and provides a fascinating description of the use of this fundamental idea within the two subject areas of lattice simulation and number theory.Complex Numbers offers a fresh and critical approach to research-based implementation of the mathematical concept of imaginary numbers. Detailed coverage includes:Riemann’s zeta function: an investigation of the non-trivial roots by Euler-Maclaurin summation.Basic theory: logarithms, indices, arithmetic and integration procedures are described.Lattice simulation: the role of complex numbers in Paul Ewald’s important work of the I 920s is analysed.Mangoldt’s study of the xi function: close attention is given to the derivation of N(T) formulae by contour integration.Analytic... calculations: used extensively to illustrate important theoretical aspects.Glossary: over 80 terms included in the text are defined.

  • Complex Numbers in n Dimensions

    • 1st Edition
    • Volume 190
    • S. Olariu
    • English
    Two distinct systems of hypercomplex numbers in n dimensions are introduced in this book, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential and trigonometric forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined.The first type of hypercomplex numbers, called polar hypercomplex numbers, is characterized by the presence in an even number of dimensions greater or equal to 4 of two polar axes, and by the presence in an odd number of dimensions of one polar axis. The other type of hypercomplex numbers exists as a distinct entity only when the number of dimensions n of the space is even, and since the position of a point is specified with the aid of n/2-1 planar angles, these numbers have been called planar hypercomplex numbers.The development of the concept of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the n-complex numbers. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and lead to the concept of n-dimensional hypercomplex residue. Expressions are given for the elementary functions of n-complex variable. In particular, the exponential function of an n-complex number is expanded in terms of functions called in this book n-dimensional cosexponential functionsof the polar and respectively planar type, which are generalizations to n dimensions of the sine, cosine and exponential functions.In the case of polar complex numbers, a polynomial can be written as a product of linear or quadratic factors, although it is interesting that several factorizations are in general possible. In the case of planar hypercomplex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible.The book presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions. The essence of this book is the interplay between the algebraic, the geometric and the analytic facets of the relations.

  • Inverse Spectral Theory

    • 1st Edition
    • Volume 130
    • Jurgen Poschel
    • English

  • Complex Analysis in Locally Convex Spaces

    • 1st Edition
    • Volume 57
    • S. Dineen
    • English

  • A First Course in Stochastic Processes

    • 2nd Edition
    • Samuel Karlin + 1 more
    • English
    The purpose, level, and style of this new edition conform to the tenets set forth in the original preface. The authors continue with their tack of developing simultaneously theory and applications, intertwined so that they refurbish and elucidate each other. The authors have made three main kinds of changes. First, they have enlarged on the topics treated in the first edition. Second, they have added many exercises and problems at the end of each chapter. Third, and most important, they have supplied, in new chapters, broad introductory discussions of several classes of stochastic processes not dealt with in the first edition, notably martingales, renewal and fluctuation phenomena associated with random sums, stationary stochastic processes, and diffusion theory.