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book Mobk070 March 22, 2007 11:7
COMPLEX VARIABLES AND THE LAPLACE INVERSION INTEGRAL 119
FIGURE 7.6: A doubly connected region
FIGURE 7.7: Derivation of Cauchy’s integral formula
the curve C 1 as shown in Fig. 7.7. Thus we can write
f (z) f (z)
dz − dz = 0 (7.27)
z − z 0 z − z 0
C 1 C 2
where both integrations are counterclockwise. Let r 0 now approach zero so that in the second
iθ
integral z approaches z 0 , z − z 0 = r 0 e iθ and dz = r 0 ie dθ. The second integral is as follows:
2π
f (z 0 )
iθ
r 0 ie dθ =− f (z 0 )i dθ =−2πif (z 0 )
r 0 e iθ
C 2
θ=0
Thus, Cauchy’s integral formula is
1 f (z)
f (z 0 ) = dz (7.28)
2πi C z − z 0
where the integral is taken counterclockwise and f (z) is analytic inside C.
