Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Example 6.13
Evaluate H C
dz=…z a† where C is any simple closed curve and z ˆ a is (a) outside
C, (b) inside C.
Solution: (a)Ifa is outside C, then f …z† ˆ1=…z a† is analytic everywhere inside
and on C. Hence by Cauchy's theorem
I
dz=…z a† ˆ0:
C
(b) Ifa is inside C and is a circle of radius 2 with center at z ˆ a so that is
inside C (Fig. 6.10). Then by Eq. (6.26) we have
I
I
dz=…z a† ˆ dz=…z a†:
C
Now on , jz aj ˆ", orz a ˆ "e i , then dz ˆ i"e i d, and
I Z
dz 2
z a ˆ i"e i Z
d 2
"e i ˆ i d ˆ 2i:
0
0
Cauchy's integral formulas
One of the most important consequences of Cauchy's integral theorem is what is
known as Cauchy's integral formula. It may be stated as follows.
If f(z) is analytic in a simply-connected region R, and z 0 is any
point in the interior of R which is enclosed by a simple closed curve
C, then
f …z 0 †ˆ 1 I
2i C
f …z†
z z 0
dz;
…6:27†
the integration around C being taken in the positive sense (counterclockwise).
Figure 6.10.
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