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38480 results in Cambridge Textbooks

Page 1148 of 1924

  • 5 - The Exponential Function

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 96-110

  • Summary

    The complex exponential function exp(z), sin(z), cos(z) are defined as power series, used to prove properties analogous to the real case, with new ideas such as Euler's formula exp(z) = cos(z) + i.sin(z). Defining e = exp(1) gives exp(z) = e^z. For z= x + iy, e^z= e^(x + iy) = e^x.e^(iy) = e^x(cos(y) + i sin(y)). Real properties of the exponential and the trigonometric functions are used to build the complex generalisations, linking symbolic properties to visual dynamic complex representations. These include the periodicity of exp, sine and cosine and introduction of other complex trigonometric and hyperbolic functions.

  • 4 - Differentiation

  • Ian Stewart, University of Warwick, David Tall, University of Warwick
  • Book:
    Complex Analysis
    Published online:
    06 September 2018
    Print publication:
    23 August 2018, pp 75-95

  • Summary

    Derivative of a complex function defined by analogy with the real case, shown to satisfy corresponding rules for combinations of functions. Differentiable paths. Power series are differentiable an infinite number of times in their circle of convergence unlike real functions which may be differentiable a certain number of times but not more. Examples are given of the 'bad behaviour' of real functions contrasting with the infinite differentiability of complex functions in an open connected domain.

  • 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.

Page 1148 of 1924