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The Winding Number and the Residue Theorem

In: Complex Analysis in One Variable

Author

Listed:
  • Raghavan Narasimhan

    (University of Chicago, Department of Mathematics)

  • Yves Nievergelt

    (Eastern Washington University, Department of Mathematics)

Abstract

The homotopy form of Cauchy’s theorem enables one to calculate many integrals of the form ∫ γ À; dz, whereÀ; is meromorphic and γ is a closed piecewise differentiable curve (it being assumed that the poles ofÀ; do not lie on Im(γ)). Formulae enabling one to do this include the so-called Cauchy formula (see §2, Theorem 2). It is, however, necessary to have some topological information about the location of the poles relative to γ. (To phrase it very vaguely, we must know how many times γ winds around a.) We begin with this topological material.

Suggested Citation

  • Raghavan Narasimhan & Yves Nievergelt, 2001. "The Winding Number and the Residue Theorem," Springer Books, in: Complex Analysis in One Variable, edition 0, chapter 0, pages 69-85, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0175-5_3
    DOI: 10.1007/978-1-4612-0175-5_3
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