# Cauchy's Integral Formula.

Recall that if and , (see def open ball). As we work in the plane, it can happen that this open ball is also called an open disk.

Theorem 5.4.1   Take a function analytic in the disk , a positive number such that and a point such that . Then:

where represents the circle whose center is and radius is equal to .

Another formulation of the above formula is as follows:

Example 5.4.2

Example 5.4.3   We wish to compute the integral .

The denominator vanishes at two points, and , both inside the contour. We will decompose this contour into two Jordan curves by the following way: draw the diameter of the circle which coincides with the x-axis. Denote:

• = the upper half circle together with the diameter, oriented positively.
• = the lower half circle together with the diameter, oriented positively.

Then (the variable travels'' twice on the diameter, but in opposite directions).

Therefore we have:

Noah Dana-Picard 2007-12-24