Page 62 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 6: Cauchy’s Integral Formula

Function: Suppose C is a curve (not necessarily a simple closed curve, just a curve) and suppose Notes
the function g is continuous on C (not necessarily analytic, just continuous). Let the function G be
defined by

g(s)
G(z) =  ds

C s z
for all z  C. We shall show that G is analytic.
Fundamental theorem of algebra: In other words, if p(z) is of degree at least one, there must be
at least one z for which p(z ) = 0. This is, of course, the celebrated fundamental theorem of
0
0
algebra.
6.7 Self Assessment

1. Suppose f is analytic in a region containing a simple closed contour C with the usual
positive orientation and its inside, and suppose z is inside C. Then it turns out that
0
1 f(z)

f(z )  2 i z z 0 dz.
0


C
This is the famous ..................
2. It says that if f is analytic on and inside a simple closed curve and we know the values f(z)
for every z on the .................., then we know the value for the function at every point inside
the curve—quite remarkable indeed.
3. Suppose C is a curve (not necessarily a simple closed curve, just a curve) and suppose the
function g is continuous on C (not necessarily analytic, just continuous). Let the ..................
G be defined by

G(z) =  g(s) ds ,

C s z
for all z  C. We shall show that G is analytic.

4. If f : D  C is .................. such that f(z)dz  for every closed curve in D, then f is analytic
0

C
in D.
1 1
5. Now suppose p(z)  0 for all z. Then is also bounded on the disk |z|  R. Thus,
p(z) p(z)
is a bounded entire function, and hence, by .................. , constant!
6. Suppose f is analytic on a closed domain D. Then, being continuous, |f(z)| must attain its
.................. somewhere in this domain.

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