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Page 1 of 3

Introduction to Complex Variables and Applications
  • Mark J. Ablowitz, Athanassios S. Fokas
  • Published online:
    18 August 2021
    Print publication:
    25 March 2021
  • The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painlevé equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.

  • 4 - Residue Calculus and Applications of Contour Integration

  • Mark J. Ablowitz, University of Colorado Boulder, Athanassios S. Fokas, University of Cambridge
  • Book:
    Introduction to Complex Variables and Applications
    Published online:
    18 August 2021
    Print publication:
    25 March 2021, pp 203-296

  • Summary

    In this chapter we extend Cauchy’s Theorem to cases where the integrand is not analytic, for example, when the integrand possesses isolated singular points. Each isolated singular point contributes a term proportional to what is called the residue of the singularity. This extension, called the residue theorem, is very useful in applications such as the evaluation of definite integrals of various types. The residue theorem provides a straightforward and sometimes the only method to compute these integrals. We also show how to use contour integration to compute the solutions of certain partial differential equations by the techniques of Fourier and Laplace transforms.

  • 1 - Complex Numbers and Elementary Functions

  • Mark J. Ablowitz, University of Colorado Boulder, Athanassios S. Fokas, University of Cambridge
  • Book:
    Introduction to Complex Variables and Applications
    Published online:
    18 August 2021
    Print publication:
    25 March 2021, pp 1-30

  • Summary

    This chapter introduces complex numbers, elementary complex functions, and their basic properties. It will be seen that complex numbers have a simple two-dimensional character which submits to a straightforward geometric description. While many results of real variable calculus carry over, some very important novel and useful notions appear in the calculus of complex functions. Applications to differential equations are briefly discussed as well.

  • 3 - Sequences, Series and Singularities of Complex Functions

  • Mark J. Ablowitz, University of Colorado Boulder, Athanassios S. Fokas, University of Cambridge
  • Book:
    Introduction to Complex Variables and Applications
    Published online:
    18 August 2021
    Print publication:
    25 March 2021, pp 106-202

  • Summary

    The representation of complex functions frequently requires the use of infinite series expansions. The best known are Taylor and Laurent series, which represent analytic functions in appropriate domains. Applications often require that we manipulate series by termwise differentiation and integration. These operations may be substantiated by employing the notion of uniform convergence. Series expansions break down at points or curves where the represented function is not analytic. Such locations are termed singular points or singularities of the function. The study of the singularities of analytic functions is vitally important in many applications including contour integration, differential equations in the complex plane, and conformal mappings.

Page 1 of 3