Shorter Notes: A Brief Proof of Cauchy's Integral Theorem

Abstract

A short proof of Cauchy's theorem for circuits homologous to 0 is presented. The proof uses elementary local properties of analytic functions but no additional geometric or topological arguments. The object of this note is to present a very short and transparent proof of Cauchy's theorem for circuits homologous to 0. The proof is based on simple 'local' properties of analytic functions that can be derived from Cauchy's theorem for analytic functions on a disc, and it may be compared with the treatment in Ahlfors [1, pp. 137-145]. It is apparent from this proof that this version of Cauchy's theorem is not only much more natural than the homotopic version which appears in several recent textbooks; it is also much easier to prove (contra Dieudonne [2, p. 192]). It is reasonable to argue that the concept of homotopy in connection with Cauchy's theorem is as extraneous as the notion of Jordan curve. We recall that if y is a circuit (= piecewise smooth, closed curve), and wEC does not lie on y, then the index of w with respect to y is Ind(y, w) = (2wri)-'f, (z-w)-ldz. It is easily proved that E= {wCC| Ind(y, w) =01 contains a neighbourhood of co and is open (see [1, p. 116]). In the following proof we give full references to the 'local' properties used in order to emphasize the elementary nature of the proof. CAUCHY'S THEOREM. Let D be an open subset of C and let y be a circuit in D. Suppose that y is homologous to 0 in D, i.e. each wED lies in the set E defined above. Then, for each f analytic on D: (i) f f(z)dz=0; (ii) Ind(y, w)f(w)=(2ri)-1f (z-w)-'f(z)dz for all wED not lying on y. PROOF. Consider g: D XD-?C defined by g(w, z) = (f(z) -f(w))/(z-w) for zpw and g(w, w) =f'(w). Then g is continuous, and for each fixed z, wi-*g(w, z) is analytic [1, p. 124]. Define h:C-C by h(w) Received by the editors January 18, 1971. AMS 1970 subject classifications. Primary 30X01, 30A86.