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,







It follows that  exists and is equal to  

Now with  denoting C we have




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.