Mathematical Methods for Physics and Engineering - Matematica.NET

24.9 CAUCHY’S THEOREMnamely Cauchy’s theorem, which is the cornerstone of the integral calculus ofcomplex variables.Before discussing Cauchy’s theorem, however, we note an important resultconcerning complex integrals that will be of some use later. Let us consider theintegral of a function f(z) along some path C. IfM is an upper bound on thevalue of |f(z)| on the path, i.e. |f(z)| ≤M on C, andL is the length of the path C,then∫∫∫∣ f(z) dz∣ ≤ |f(z)||dz| ≤M dl = ML. (24.39)CcCIt is straightforward to verify that this result does indeed hold for the complexintegrals considered earlier in this section.24.9 Cauchy’s theoremCauchy’s theorem states that if f(z) is an analytic function, and f ′ (z) is continuousat each point within and on a closed contour C, then∮f(z) dz =0. (24.40)CIn this statement and from now on we denote an integral around a closed contourby ∮ C .To prove this theorem we will need the two-dimensional form of the divergencetheorem, known as Green’s theorem in a plane (see section 11.3). This says thatif p and q are two functions with continuous first derivatives within and on aclosed contour C (bounding a domain R) inthexy-plane, then∫∫R( ∂p∂x + ∂q∂y)dxdy =∮C(p dy− qdx). (24.41)With f(z) =u + iv and dz = dx + idy, this can be applied to∮∮∮I = f(z) dz = (u dx− vdy)+i (v dx+ udy)to give∫∫ [ ∂(−u)I =∂yRCC+ ∂(−v) ] ∫∫ [ ∂(−v)dx dy + i+ ∂u ]dx dy. (24.42)∂xR ∂y ∂xNow, recalling that f(z) is analytic and therefore that the Cauchy–Riemannrelations (24.5) apply, we see that each integrand is identically zero and thus I isalso zero; this proves Cauchy’s theorem.In fact, the conditions of the above proof are more stringent than they needbe. The continuity of f ′ (z) is not necessary for the proof of Cauchy’s theorem,849C