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38480 results in Cambridge Textbooks

Page 1147 of 1924

  • 16 - Homology Version of Cauchy’s Theorem

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 350-373

  • Summary

    This chapter extends the homotopy versions of Cauchy's Theorem in Chapter 9 to consider how 'holes' in the domain caused by singularities affect complex integrals that wind round them. It uses a 'bare hands' approach based on step paths guaranteed by the Paving Lemma. A sum of closed rectangles in the domain is a cycle; if the interior of each rectangle also lies in the domain, the cycle is called a boundary. The homology version of Cauchy's Theorem shows that the integral round a boundary is zero.

  • 17 - The Road Goes Ever On …

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 374-381

  • Summary

    The final chapter looks to the evolving theory of complex analysis, including references to the Riemann Hypothesis, modular functions, functions of several complex variables, complex manifolds, complex dynamics and fractal geometry.

  • 6 - Integration

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 111-148

  • Summary

    The subtleties of complex integration are discussed in detail. Length is defined for a contour (piece-wise smooth path). The integral of a differentiable function along a path is defined for a continuous function. The Fundamental Theorem of Contour integration is proved for a function f in a domain D with an antiderivative F (meaning F' = f throughout D). The Estimation Lemma is proved to estimate the size of a contour integral of a function f in terms of the length of the contour and the maximum value of |f|. This is a foundational concept for later proofs.

  • 14 - Analytic Continuation

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 289-314

  • Summary

    A power series is limited to a disk; an analytic function may be naturally extended through a sequence of overlapping domains to give an analytic continuation where different sequences may give different extensions. This leads to a multiform function that may be represented as a Riemann surface. Examples include the logarithm, square root, complex powers, conformal functions. General properties include conformal maps, contour integrals on Riemann surfaces.

  • 1 - Algebra of the Complex Plane

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 13-23

  • Summary

    Complex numbers are introduced algebraically as ordered pairs of real numbers (x,y) with a formal definition of addition and multiplication. They are represented geometrically as points in the plane with (x,y) also denoted by x+iy and multiplication corresponds to defining i squared as –1. Topics include real and imaginary parts, modulus, complex conjugate, polar coordinates, with a proof that the complex numbers cannot be ordered.

Page 1147 of 1924