Cauchy integral theorem
Let be an open, simply connected domain, and a finite subset. Let be a function whose complex derivative exists for all , and that is bounded near all . Let be a closed contour that avoids the exceptional points. Then, the integral of around 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 exist and be continuous. The existence of implies the Cauchy-Riemann equations, which in turn can be restated as the fact that the complex-valued differential 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 , 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 century E. Goursat found a proof of the integral theorem that merely required that 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 be an open domain containing a rectangle
If the complex derivative of a function exists at all points of , then the contour integral of around the boundary of vanishes; in symbols
Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.
|Title||Cauchy integral theorem|
|Date of creation||2013-03-22 12:54:12|
|Last modified on||2013-03-22 12:54:12|
|Last modified by||rmilson (146)|
|Synonym||fundamental theorem of complex analysis|