Abstract
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.
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© 2012 Springer-Verlag London
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Barreira, L., Valls, C. (2012). Holomorphic Functions. In: Complex Analysis and Differential Equations. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-4008-5_2
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DOI: https://doi.org/10.1007/978-1-4471-4008-5_2
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4007-8
Online ISBN: 978-1-4471-4008-5
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