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THE CAUCHY INTEGRAL THEOREM (^145)
imag z-plane
n-
u
Figure 6. 5 The Closed Cauchy Integral Path
plane. Consider an integral
along a closed contour C which lies entirely within the analytic region, as shown in
Figure 6.5. Because ~ ( g )= u(x, y ) + i v ( x , y ) and dg = d x + i d y , Equation 6.63 can
be rewritten as
Stokes’s theorem
can now be applied to both integrals on the RHS of Equation 6.64. For a closed line integral confined to the xy-plane, Equation 6.65 becomes
The first integral on the RHS of Equation 6.64 is handled by identifying V, = u(x, y ) ,
v = - v ( x , y ) , and S as the surface enclosed by C , so
146 INTRODUCTION TO COMPLEX VARIABLES
The second integral of Equation 6.64 is handled in a similar manner, by identifying
V, = v ( x , y ) and V, = u(x, y ) , so
v(x, y ) + d y u(x, y ) ] = d x d y - - -.
1 (2 :;)
Because &) is analytic and obeys the Cauchy-Riemann conditions inside the entire closed contour, &/ax = dv/@ and & / d y = - d v / d x , and we are left with the elegant result that
6.5 CONTOUR DEFORMATION
There is an immediate, practical consequence of the Cauchy Integral Theorem. The contour of an complex integral can be arbitrarily deformed through an analytic region without changing the value of the integral. Consider the integral
along the contour C, from a to b, drawn as the solid line in Figure 6.6(a). If we add to this an integral with the same integrand, but around the closed contour C , shown as the dotted line in Figure 6.6(a), the result will still be I, as long as the integrand is analytic in the entire region inside C:
~ imag imag
real real i
Figure 6. 6 Contour Deformation
