The Cauchy-Goursat Theorem and Contour Integrals

[Bachelier]

The Cauchy-Goursat Theorem states:

Let ##f## be holomorphic in a simply connected domain D. If C is a simple closed contour that lies in D, then

##\int_C f(z) \mathrm{d}t = 0 ##​

Now if ##f## is holo just on ##|C| \bigcup \ int(C)## (i.e ##f## holo only on the contour and inside of it, if we take ##z_0 \in \mathbb{C}## can we deduce from the theorem that

## \int_C \frac{f'(z)}{z-z_0} \mathrm{d}t = \int_C \frac{f(z)}{(z-z_0)^2} \mathrm{d}t = 0 ## whether ##z_0 \in \left[\ |C| \bigcup \ int(C) \right]## or not​

Since both equal ##0##.

Also where does ##z## must belong to for the theorem to apply?

 

[Bachelier]

I think I got it. I cannot apply Cauchy inside because of the singularity problem.

Now my problem is, even though we don't know if ##f## is holo outside by the givens, can we still apply Cauchy outside ##C## and claim that

[tex]\int_c{f(z)} \mathrm{d}t = 0[/tex]​