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Conformal Mappings

  • John M. Howie
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

This chapter explores the consequence of a remarkable geometric property of holomorphic functions. Look again at Figures 3.1 and 3.2 on page 42. For arbitrary k and l the lines u = k and v = l in the w-plane are of course mutually perpendicular, and visually at least it seems that the corresponding hyperbolic curves x2y2 = k and 2xy = l in the z-plane are also perpendicular. Again, the lines x = k and y = l are mutually perpendicular, and it appears also that the corresponding parabolic curves in the w-plane are also perpendicular. These observations are in fact mathematically correct (see Exercise 11.1), and are instances of a general theorem to be proved shortly. First, however, we need to develop a little more of the theory of the parametric representation of curves that was introduced in Section 5.2.

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Notes

  1. 1.
    Pierre Simon Laplace, 1749–1827.Google Scholar
  2. 2.
    Johann Peter Gustav Lejeune Dirichlet, 1805–1859.Google Scholar
  3. 3.
    August Ferdinand Möbius 1790–1868.Google Scholar
  4. 4.
    Nikolai Egorovich Joukowski (Zhukovskii), 1847–1921.Google Scholar

Copyright information

© Springer-Verlag London 2003

Authors and Affiliations

  • John M. Howie
    • 1
  1. 1.School of Mathematics and Statistics, Mathematical InstituteUniversity of St AndrewsNorth Haugh, St Andrews, FifeUK

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