Cauchy integral formula again
July 20, 2012 Leave a comment
We’ve already proved the Cauchy integral formula:
Theorem 1 (Cauchy integral formula) For , , and any counterclockwise circle in about (or any loop homotopy equivalent to it in ),
It’s really quite amazing. Thinking about it slightly more algebraically, consider the operator
Yes, this looks complicated, but the idea is you plug in a function , and it spits out another function depending on which you obtain by integrating that thing in a loop around the input. You shouldn’t expect this operator to behave in any reasonable way, but if you plug in a holomorphic function , it spits out . That is,
Theorem 2 (Cauchy integral formula) Holomorphic functions are eigenfunctions of the above operator with eigenvalue .
Thank you to Jordy Greenblatt for pointing this out to me. This result is what makes complex analysis so nice. Remember how eigenvectors were really nice for linear tranformations. This is the basically the same thing.
Theorem 3 (Cauchy integral formula) Let , and , and a loop about , as above. Then
Proof: Expand as a power series about . Then
Taking the th derivative and evaluating at yields
Recalling from our proof that holomorphic functions are analytic,