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Cauchy integral theorem
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(Theorem)
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We also have the following, technically important generalization involving removable singularities.
Theorem 2 Let $U\subset\cnums$ be an open, simply connected domain, and $S\subset U$ a finite subset. Let $f:U\backslash S \rightarrow \cnums$ be a function whose complex derivative exists for all $z\in U\backslash S$ , and that is bounded near all $z\in S$ . Let $\gamma\subset U\backslash S$ be a closed contour that avoids the exceptional points. Then, the integral of $f$ around $\gamma$ vanishes.
Cauchy's theorem is an essential stepping stone in the theory of complex analysis. It is required for the proof of the Cauchy integral formula, which in turn is required for the proof that the existence of a complex derivative implies a power series representation.
The original version of the theorem, as stated by Cauchy in the early 1800s, requires that the derivative $f'(z)$ exist and be continuous. The existence of $f'(z)$ implies the Cauchy-Riemann equations, which in turn can be restated as the fact that the complex-valued differential $f(z)\, dz$ is closed. The original proof makes use of this fact, and calls on Green's Theorem to conclude that the contour integral vanishes. The proof of Green's theorem, however, involves an interchange of order in a double integral, and this can only be justified if the integrand, which involves the real and imaginary parts of $f'(z)$ , is assumed to be continuous. To this date, many authors prove the theorem
this way, but erroneously fail to mention the continuity assumption.
In the latter part of the $19\supth$ century E. Goursat found a proof of the integral theorem that merely required that $f'(z)$ exist. Continuity of the derivative, as well as the existence of all higher derivatives, then follows as a consequence of the Cauchy integral formula. Not only is Goursat's version a sharper result, but it is also more elementary and self-contained, in that sense that it is does not require Green's theorem. Goursat's argument makes use of rectangular contour (many authors use triangles though), but the extension to an arbitrary
simply-connected domain is relatively straight-forward.
Theorem 3 (Goursat) Let $U$ be an open domain containing a rectangle $$R = \{ x+iy\in\cnums: a\leq x\leq b\,, c\leq y\leq d\}.$$ If the complex derivative of a function $f:U\rightarrow \cnums$ exists at all points of $U$ , then the contour integral of $f$ around the boundary of $R$ vanishes; in symbols $$\oint_{\partial R} f(z)\,dz = 0.$$
- Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.
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"Cauchy integral theorem" is owned by rmilson. [ full author list (2) ]
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Cross-references: boundary, rectangle, extension, triangles, argument, consequence, imaginary parts, real, integrand, double integral, order, proof of Green's theorem, contour integral, Green's theorem, Cauchy-Riemann equations, continuous, derivative, theorem, representation, power series, implies, Cauchy integral formula, proof, complex analysis, theory, Cauchy's theorem, integral, points, near, bounded, subset, finite, removable singularities, vanishes, contour, closed, complex derivative, function, domain, simply connected, open
There are 10 references to this entry.
This is version 13 of Cauchy integral theorem, born on 2002-08-01, modified 2008-09-04.
Object id is 3253, canonical name is CauchyIntegralTheorem.
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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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