So I am just looking at some exercise to prepare myself for the exam and just seem confused when I can apply these Theorems: 

> 
> a) Let $A = 1 + i, B = −1 + i, C = −1 − i, D = 1 − i$ and γ be the path [A, B, C, D, A], that is the
>  contour of the square ABCD.
> 
> 
> Compute the path Integrals $\int\_{γ} \frac{e^{z}-e^{-z}}{e^{z}-4}$
>  and $\int\_{γ} \frac{e^{z}}{{z}^4}$
> 
> 
> 

So to the first Integral I know, that the function $f(z) = \frac{e^{z}-e^{-z}}{e^{z}-4}$ is holomorphic and thus continous on the entire Complex plane and γ is a closed path. 

> 
> One of my Theorem only states, that if the domain is star-shaped and f is holomorphic on that domain, then f has an antiderivative. And if this is true and we are integrating over a closed curve, the Integral must be equal 0. 
> 
> 
> 

So why can I apply this to this example, or is there an easier way of doing it?

The second Integral can be solved with Cauchy's Integral formula for derivatives, but I am also not sure why I can apply it here?

Some explanation would help me a lot :)