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38482 results in Cambridge Textbooks

Page 1149 of 1925

  • 2 - Topology 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 24-58

  • Summary

    Looks at simple properties of plane topology, including open and closed sets, limits and continuity. Paths are continuous maps from a real interval to the complex plane. A subset S of the plane is (path-) connected if any two points in S can be connected by a path in S. A domain is an open connected subset. To simplify proofs, the Paving Lemma is proved, showing that a path in a domain D can be covered by a finite number of discs so that any two points within D can be joined by a step-path in D consisting of horizontal and vertical steps. This allows visual ideas of continuous paths to be interpreted using only horizontal and vertical steps, linking intuition in the complex plane to the real case.

  • 3 - Power 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 59-74

  • Summary

    Generalising convergence of real sequences and series to the complex case, Cauchy's General Principle of Convergence for sequences, absolute convergence for series, comparison test, and ratio test. Complex power series and absolute convergence in a disc. Power series are fundamental in the study of differentiable complex functions.

  • 9 - Homotopy Versions 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 187-206

  • Summary

    Integration of analytic functions along paths. Homotopy, with fixed endpoints, for closed paths. Cauchy's Theorem for a boundary, for homotopic paths with fixed endpoints and for homotopic closed paths. Various versions of Cauchy's Theorem all build on the local existence of an antiderivative and the Fundamental Theorem of Contour Integration.

  • 8 - 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 169-186

  • Summary

    Cauchy's Theorem proved for a triangle, a star domain, for any path that does not wind round a point outside the domain of the function, using local antiderivative and integration around nearby step-path. Discussion of cuts, Jordan contours, simply connected domains.

Page 1149 of 1925