How do i define $f(z_0)$ in an contour integral?

Question

I have a Problem described as following: $$ \oint \dfrac{\frac{3}{7}}{z+3}+\dfrac{\frac{4}{7}}{z-4}\,\mathrm{dz} \;\;\text{ on the contour } |z-3|=2 $$ and i have come as far to use the formulars below, but i am not sure how to define $f(z_0)$
$$ f(z_0)=\dfrac{1}{2\pi i} \oint \dfrac{f(z)}{z-z_0},\mathrm{dz} = f(z_0)\cdot 2\pi i $$ $z_0=4$ and $z_0=-3$ but only $z_0=4$ is inside the contour and is the only part i look at.

is $$f(z_0)=\frac{4}{7}\frac{1}{z-4}$$ if not how do i find it?

hope someone can help.

Answer 1

Since the only point at which $\frac{3/7}{z+3}$ is undefined is outside the region bounded by the given contour, $\oint\frac{3/7}{z+3}\,\mathrm dz=0$. And so\begin{align}\oint\frac{3/7}{z+3}+\frac{4/7}{z-4}\,\mathrm dz&=\oint\frac{4/7}{z-4}\,\mathrm dz\\&=2\pi i\times\frac47\\&=\frac{8\pi i}7.\end{align}