PlanetMath: Cauchy integral formula

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Cauchy integral formula

(Theorem)

The formulas.

Let $D=\{z\in\mathbb{C} : \vert z-z_0\vert <R\}$ be an open disk in the complex plane, and let

be a holomorphic ¹ function defined on some open domain that contains

and its boundary. Then, for every $z\in D$ we have

$\displaystyle f(z)$	$\displaystyle =$	$\displaystyle \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta-z}\ d\zeta$
$\displaystyle f'(z)$	$\displaystyle =$	$\displaystyle \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{(\zeta-z)^2}\ d\zeta$
	$\displaystyle \vdots$
$\displaystyle f^{(n)}(z)$	$\displaystyle =$	$\displaystyle \frac{n!}{2 \pi i} \oint_C \frac{f(\zeta)}{(\zeta-z)^{n+1}}\ d\zeta$

Here $C=\partial D$ is the corresponding circular boundary contour, oriented counterclockwise, with the most obvious parameterization given by

$\displaystyle \zeta=z_0+R e^{it},\quad 0\leq t\leq 2\pi.$

Discussion.

The first of the above formulas underscores the “rigidity” of holomorphic functions. Indeed, the values of the holomorphic function inside a disk

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.

Generalization.

The following technical generalization of the formula is needed for the treatment of removable singularities. Let

be a finite subset of

, and suppose that

is holomorphic for all $z\notin S$ , but also that

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 is any domain and is any closed rectifiable curve in ; in this case, the formula becomes

$\displaystyle \eta(C,z) f(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta-z}\ d\zeta$

where $\eta(C,z)$ denotes the winding number of

. It is valid for all points $z \in D \setminus S$ which are not on the curve

Footnotes

...¹: 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.

"Cauchy integral formula" is owned by djao. [ full author list (3) | owner history (1) ]

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Attachments:: proof of Cauchy integral formula (Proof) by rmilson
Cauchy integral formula in several variables (Theorem) by jirka

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Cross-references: curve, points, winding number, rectifiable curve, closed, Cauchy residue theorem, near, bounded, subset, finite, removable singularities, limit, outcome, integral, terms, derivative, obvious, oriented, contour, circular, boundary, contains, domain, function, hypothesis, proof, equivalent, power series, representable, analytic functions, complex derivative, necessary, holomorphic, complex plane, open
There are 14 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 12623 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)

Pending Errata and Addenda

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