Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESyBC 1RC 2AxFigure 24.10 Two paths C 1 and C 2 enclosing a region R.analyticity of f(z) within and on C being sufficient. However, the proof thenbecomes more complicated and is too long to be given here. §The connection between Cauchy’s theorem and the zero value of the integralof z −1 around the composite path C 4 discussed towards the end of the previoussection is apparent: the function z −1 is analytic in the two regions of the z-planeenclosed by contours (C 2 and C 3a )and(C 2 and C 3b ).◮Suppose two points A and B in the complex plane are joined by two different paths C 1and C 2 . Show that if f(z) is an analytic function on each path and in the region enclosedby the two paths, then the integral of f(z) is the same along C 1 and C 2 .The situation is shown in figure 24.10. Since f(z) isanalyticinR, it follows from Cauchy’stheorem that we have∫ ∫∮f(z) dz − f(z) dz = f(z) dz =0,C 1 C 2 C 1 −C 2since C 1 − C 2 forms a closed contour enclosing R. Thus we immediately obtain∫∫f(z) dz = f(z) dz,C 1 C 2and so the values of the integrals along C 1 and C 2 are equal. ◭An important application of Cauchy’s theorem is in proving that, in somecases, it is possible to deform a closed contour C into another contour γ in sucha way that the integrals of a function f(z) around each of the contours have thesame value.§ The reader may refer to almost any book that is devoted to complex variables and the theory offunctions.850