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38480 results in Cambridge Textbooks

Page 1146 of 1924

  • 10 - Taylor Series

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 207-224

  • Summary

    Taylor series is proved using Cauchy's Integral Formula which is itself proved using Cauchy's Theorem and the Estimation Lemma. Morera's Theorem (an inverse of Cauchy's Theorem), Cauchy's Estimate for the remainder in Taylor's series, Liouville's Theorem and a proof of the Fundamental Theorem of Algebra. Also studied: zeros of functions, extending functions to a larger domain, local maxima and minima of the modulus of a function.

  • 12 - Residues

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 243-267

  • Summary

    Cauchy's Residue Theorem and its use for calculating certain types of infinite integral, infinite series, and a range of other exercises.

  • 15 - Infinitesimals in Real and Complex Analysis

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 315-349

  • Summary

    Infinitesimal quantities arose in history and continue to be used as 'arbitrarily small quantities' in applied mathematics. This chapter provides a formal foundation for an extension of the real and complex numbers containing infinitesimals that can be manipulated algebraically and visualised dynamically by magnification. This relates to classical theory, modern standard analysis and non-standard analysis. This gives a formal theory to visualise infinitesimal figures rotated and scaled in conformal transformations in the plane and on Riemann surfaces.

  • 0 - The Origins of Complex Analysis, and Its Challenge to Intuition

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 1-12

  • Summary

    This chapter looks at the historical origins of complex numbers, controversies over their meaning, the development of a visual representation in the plane, the nineteenth century development of complex analysis where Cauchy used infinitesimals, and the arithmetization of analysis by Weiestrass leading to modern formal analysis which was the basis of the first edition. This second edition offers a new formal approach in an extension field which includes infinitesimals both algebraically and geometrically, linking to the historical use of infinitesimals still found in modern applied mathematics and to non-standard analysis. This is considered in the nature of human evolution of ideas relating to discovery and invention and the relationship between visual intuition and mathematical rigour.

Page 1146 of 1924