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Cauchy integral formula
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(Theorem)
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Let $D=\{z\in\mathbb{C} : |z-z_0| <R\}$ be an open disk in the complex plane, and let $f(z)$ be a holomorphic 1 function defined on some open domain that contains $D$ and its boundary. Then, for every $z\in D$ we have \begin{eqnarray*} f(z) &=& \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta-z}\ d\zeta \\ f'(z) &=& \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{(\zeta-z)^2}\ d\zeta \\ & \vdots & \\ f^{(n)}(z) &=& \frac{n!}{2 \pi i} \oint_C \frac{f(\zeta)}{(\zeta-z)^{n+1}}\ d\zeta \end{eqnarray*}Here $C=\partial D$ is the corresponding circular boundary contour, oriented counterclockwise, with the most obvious parameterization given by $$\zeta=z_0+R e^{it},\quad 0\leq t\leq 2\pi.$$
The first of the above formulas underscores the ``rigidity'' of holomorphic functions. Indeed, the values of the holomorphic function inside a disk $D$ are completely specified by its values on the boundary of the disk. The second formula is useful, because it gives the derivative in terms of an integral, rather than as the outcome of a limit process.
The following technical generalization of the formula is needed for the treatment of removable singularities. Let $S$ be a finite subset of $D$ , and suppose that $f(z)$ is holomorphic for all $z\notin S$ , but also that $f(z)$ is bounded near all $z\in S$ . Then, the above formulas are valid for all $z\in D\setminus S$ .
Using the Cauchy residue theorem, one can further generalize the integral formula to the situation where $D$ is any domain and $C$ is any closed rectifiable curve in $D$ ; in this case, the formula becomes $$ \eta(C,z) f(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta-z}\ d\zeta $$ where $\eta(C,z)$ denotes the winding number of $C$ . It is valid for all points $z \in D \setminus S$ which are not on the curve $C$ .
Footnotes
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- It is necessary to draw a distinction between holomorphic functions (those having a complex derivative) and analytic functions (those representable by power series). The two concepts are, in fact, equivalent, but the standard proof of this fact uses the Cauchy Integral Formula with the (apparently) weaker
holomorphicity hypothesis.
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"Cauchy integral formula" is owned by djao. [ full author list (3) | owner history (1) ]
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Cross-references: curve, points, winding number, rectifiable curve, closed, Cauchy residue theorem, valid, near, bounded, subset, finite, removable singularities, limit, outcome, integral, terms, derivative, formulas, obvious, oriented, contour, circular, boundary, contains, domain, open, function, hypothesis, proof, equivalent, power series, representable, analytic functions, complex derivative, necessary, holomorphic, complex plane, open disk
There are 12 references to this entry.
This is version 21 of Cauchy integral formula, born on 2001-12-28, modified 2004-03-15.
Object id is 1150, canonical name is CauchyIntegralFormula.
Accessed 15011 times total.
Classification:
| AMS MSC: | 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions) |
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Pending Errata and Addenda
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