Homotopy and Homology version of Cauchy's Theorem

1. Apr 29, 2014

Maybe_Memorie

1. The problem statement, all variables and given/known data

"Give examples showing no assumption in your statement of Cauchy's theorem can be removed. Justify your answer."

2. Relevant equations

I'm quite poor with LaTeX so I've providing links to the statement and proof of each version of the theorem.

Homotopy Version
Homology Version

3. The attempt at a solution

This is quite important as it is asked on every Complex Analysis exam my lecturer has set. However there's actually nothing in the notes regarding this. I'll start with the Homology version since the statement is shorter.

All we're assuming is that f is holomorphic in a subset of C and the integration is done on a 1-cycle.

Suppose f=1/z, and the path is a unit circle who's interior contains 0. Then the value of the integral around the path is 2∏i =/= 0, which it should by Cauchy. So f needs to be holomorphic in the interior. However I don't know how to write paths as cycles, so don't know how to finish this example...

The other assumption is that the curve is a 1-cycle. If it isn't a 1-cycle then the path isn't closed. Again, my lack of knowledge of cycles is causing issues here...

For the homotopy version, the holomorphic part is the same as for the last one, but the homotopy part is causing me problems.

I'd appreciate any help.

2. Apr 30, 2014

Maybe_Memorie

Can anyone help?