proof of Cauchy integral formula
Let be a disk in the complex plane, a finite subset, and an open domain that contains the closed disk . Suppose that
Hence, by a straightforward compactness argument we also have that is bounded on , and hence bounded on .
Let be given, and set
where . Note that is holomorphic and bounded on . The second assertion is true, because
Therefore, by the Cauchy integral theorem
where is the counterclockwise circular contour parameterized by
If is such that , then
The proof is a fun exercise in elementary integral calculus, an application of the half-angle trigonometric substitutions.
Thanks to the Lemma, the right hand side of (1) evaluates to Dividing through by , we obtain
|Title||proof of Cauchy integral formula|
|Date of creation||2013-03-22 12:47:23|
|Last modified on||2013-03-22 12:47:23|
|Last modified by||rmilson (146)|