Page 128 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

118 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
On the other hand, if we perform the same integral along the x axis to x = 2 and then
along the vertical line x = 2 to the same point, path OAB in Fig. 7.5, we find that

2 1 1
8 2 11
2 2 2
x dx + (2 + iy) idy = + i (4 − y + 4iy)dy = + i
3 3 3
x=0 y=0 y=0
2
This happened because the function z is analytic within the region between the two curves.
In general, if a function is analytic in the region contained between the curves, the integral

f (z)dz (7.23)

C
is independent of the path of C. Since any two integrals are the same, and since if we integrate
the first integral along BO only the sign changes, we see that the integral around the closed
contour is zero.

f (z)dz = 0 (7.24)
C
This is called the Cauchy–Goursat theorem and is true as long as the region R within the
closed curve C is simply connected and the function is analytic everywhere within the region. A
simply connected region R is one in which every closed curve within it encloses only points in R.
The theorem can be extended to allow for multiply connected regions. Fig. 7.6 shows
a doubly connected region. The method is to make a cut through part of the region and to
integrate counterclockwise around C 1 , along the path C 2 through the region, clockwise around
the interior curve C 3 , and back out along C 4 . Clearly, the integral along C 2 and C 4 cancels, so
that

f (z)dz + f (z)dz = 0 (7.25)
C 1 C 3
where the first integral is counterclockwise and second clockwise.

7.1.2 The Cauchy Integral Formula
Now consider the following integral:

f (z)dz

(7.26)
(z − z 0 )
C
If the function f (z) is analytic then the integrand is also analytic at all points except z = z 0 .
We now form a circle C 2 of radius r 0 around the point z = z 0 that is small enough to fit inside