6.6 • THE THEOREMS OF MORERA AND L!OUVILLE, AND EXTENSIONS 241Use Cauchy's integral formula to show that1 r (e- 1rd~
P.,. (z) = 2-iri f c 2" (~ - z)"+^1 'where C is a simple closed contour having positive orientation and z lies inside C.- Discuss the importance of being able to define an analytic function f (z) with
the contour integral in Formula (6-44). How does this definition differ from other
definitions of a function that you have learned?
6.6 THE THEOREMS OF MORERA
AND LIOUVILLE, AND EXTENSIONS
In this section we investigate some of the qualitative properties of analytic and
harmonic functions. Our first res1,!1t shows that the existence of an antiderivative
for a continuous function is equivalent to the statement that the integral of f is
independent of the path of integration. This result is stated in a form that will
serve as a converse of the Cauchy- Goursat theorem.
Cauchy's integral formula shows how the value f (zo) can be represented by
a certain contour integral. If we choose the contour of integration C to be a circle