Holomorphic Functions

  • Luis BarreiraEmail author
  • Claudia Valls
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


In this chapter we introduce the notion of a differentiable function, or of a holomorphic function. It turns out that differentiability is characterized by a pair of (partial differential) equations—the Cauchy–Riemann equations. We also introduce the notion of the integral along a path and we study its relation to the notion of a holomorphic function. Finally, we introduce the index of a closed path, we obtain Cauchy’s integral formula for a holomorphic function, and we discuss the relation between integrals and homotopy.

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal

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