Page 61 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes  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 function G
be defined by

g(s)
G(z) =  ds ,

C s z
for all z  C. We shall show that G is analytic.
Suppose f is analytic in a region D and suppose C is a positively oriented simple closed

curve in D. Suppose also the inside of C is in D. Then from the Cauchy Integral formula, we
know that

f(s)

2 if(z)   ds

C s z
and so with g = f in the formulas just derived, we have
1 f(s) 2 f(s)
f'(z)   ds, and f''(z)   ds
2pi (s z) 2 2 i (s z) 3



C
C
for all z inside the closed curve C. Meditate on these results. They say that the derivative
of an analytic function is also analytic. Now suppose f is continuous on a domain D in
which every point of D is an interior point and suppose that f(z)dz  , for every closed
0

C
curve in D.
Suppose f is entire and bounded; that is, f is analytic in the entire plane and there is a

constant M such that |f(z)|  M for all z. Then it must be true that f’(z) = 0, identically. To
M
see this, suppose that f’(w)  0, for some w. Choose R large enough to insure that  f'(w) .
R
Now let C be a circle centered at 0 and with radius  > max{R, |w|}. Then we have :
M  1 f(s) ds

R f'(w)  2 i (s w) 2


C
1 M M
 2  ,
2  2 
a contradiction. It must therefore be true that there is no w for which f’(w)  0; or, in other
words, f’(z) = 0 for all z. This, of course, means that f is a constant function. What we have
shown has a name, Liouville’s Theorem:
The only bounded entire functions are the constant functions.

6.6 Keywords

Cauchy Integral Formula: 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
0
that
1 f(z)

f(z )  2 i z z 0 dz.
0


C
This is the famous Cauchy Integral Formula.
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