Mathematical Methods for Physics and Engineering - Matematica.NET

24.10 CAUCHY’S INTEGRAL FORMULAyCC 1γC 2Figure 24.11 The contour used to prove the result (24.43).x◮Consider two closed contours C and γ in the Argand diagram, γ being sufficiently smallthat it lies completely within C. Show that if the function f(z) is analytic in the regionbetween the two contours then∮∮f(z) dz = f(z) dz. (24.43)CγTo prove this result we consider a contour as shown in figure 24.11. The two close parallellines C 1 and C 2 join γ and C, which are ‘cut’ to accommodate them. The new contour Γso formed consists of C, C 1 , γ and C 2 .Within the area bounded by Γ, the function f(z) is analytic, and therefore, by Cauchy’stheorem (24.40),∮f(z) dz =0. (24.44)ΓNow the parts C 1 and C 2 of Γ are traversed in opposite directions, and in the limit lie ontop of each other, and so their contributions to (24.44) cancel. Thus∮ ∮f(z) dz + f(z) dz =0. (24.45)CThe sense of the integral round γ is opposite to the conventional (anticlockwise) one, andso by traversing γ in the usual sense, we establish the result (24.43). ◭A sort of converse of Cauchy’s theorem is known as Morera’s theorem, whichstates that if f(z) is a continuous function of z in a closed domain R bounded byacurveC and, further, ∮ Cf(z) dz =0,thenf(z) is analytic in R.γ24.10 Cauchy’s integral formulaAnother very important theorem in the theory of complex variables is Cauchy’sintegral formula, which states that if f(z) is analytic within and on a closed851