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9.1 Cauchys Theorem for a Triangle

Theorem 1. (Cauchys theorem for a triangle). Let be analytic in a domain and let be continuous on If C is a triangle in , then Proof. By Greens Theorem (exercise 1): where is the inside of the triangle C

Put Since is analytic on , and satisfy the

Cauchy-Riemann equations and thus Remarks. Indeed, the hypothesis that is continuous on is not needed and there are other technical proofs. The hypotheses in Greens Theorem are satisfied under the assumption that is continuous on 9.2  Cauchys Theorem for a star-like domain

Theorem 2. (Cauchys theorem). If is analytic in a star-like domain and  is a closed contour in , then Proof. Since is a star-like domain, there is such that the segment between and is in for each Define where the path of integration is that segment. Let be a point in off this line segment.
By continuity of at , given we choose small enough so that Now let be small enough and satisfying Consider the triangle with vertices Then, by Cauchys Theorem for a triangle, Hence Then It follows that exists and is equal to Now with denoting C we have Thus since C is closed.

9.3 Cauchys Integral Theorem

Theorem 3. (Cauchys Integral Theorem). Let be analytic in a disk and C is the circle . Then for every in the disk we have Proof. Consider the function if and if .

Clearly, g is analytic in the star-like domain and by hypothesis C is inside .  Thus by Cauchys Theorem ; that is, or and the result follows.

9.4 Cauchys Integral Formula

Theorem 4. Proof.  Differentiate  Cauchys Integral Formula n times with respect to  z. 