Applications of the Cauchy Theory

Steven G. Krantz
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Steven G. Krantz: Washington University in St. Louis, Department of Mathematics

Chapter Chapter 3 in Handbook of Complex Variables, 1999, pp 31-39 from Springer

Abstract: Abstract Let $$U \subseteq \mathbb{C}$$ be an open set and let f be holomorphic on U. Then C∞(U). Moreover, if $$\bar{D}(P,r) \subseteq U$$ and z ∈ D(P,r), then 3.1.1.1 $$\begin{array}{*{20}{c}} {{{{\left( {\frac{\partial }{{\partial z}}} \right)}}^{k}}f(z) = \frac{{k!}}{{2\pi i}}\oint_{{|\zeta - P| = r}} {\frac{{f(\zeta )}}{{{{{\left( {\zeta - z} \right)}}^{{k + 1}}}}}d\zeta ,} } & {k = 0,1,2, \ldots .} \\ \end{array}$$

Date: 1999
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DOI: 10.1007/978-1-4612-1588-2_3

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