Winding number of $f$ about an arbitrary point $w_0$ The argument principle tells us given some assumptions of the function $f$ and the contour $\gamma$, we have the winding number of $f$ along $\gamma$ with respect to $0$, $$W(f,\gamma, 0) = Z-P,$$ where $Z$ is the number of the zeros of $f$ and $P$ is the number of poles of $f$ in int$(\gamma)$. 
For an arbitrary point $w_0 \in \mathbb C$, do we use the same formula $W(f,\gamma, w_0) = Z-P$ with the restriction that $f(z) \not = w_0$ for $z\in \gamma$?

The winding number $W(f, \gamma, w_0)$ is defined as
$$ W(f, \gamma, w_0) = \oint_{f \circ \gamma} \frac {1} {w - w_0} dw = \oint_\gamma  \frac{f'(z)}{f(z) - w_0 } dz .$$
From this definition, it is clear that
$$ W(f, \gamma, w_0) = W(f - w_0, \gamma , 0).$$
So $W(f, \gamma, w_0)$ counts the number of zeroes minus the number of poles of the function $$z \mapsto f(z) - w_0$$ in the interior of $\gamma$.
