Show that the image of a simple contour under $f(z) = \dfrac{1}{z-1}$ encircles the origin once counter clockwise

Question

Let $$f(z) = \dfrac{1}{z-1}, z \in \mathbb{C}$$

Let $\mathcal{D}$ be a simple contour (e.g. no self loop, etc.) that encircles $z = 1$ once clockwise.

Then claim: $f(\mathcal{D})$ is a contour that encircles the origin $z = 0$ once counter-clockwise

Can someone explain why this is so?

My attempt is take $z = re^{i\theta}$, and suppose that $z \in \mathcal{D}$

Then $f(z) = \dfrac{1}{ re^{i\theta}-1} = \dfrac{1}{ re^{i\theta}-e^{i\theta}e^{-i\theta}} = \dfrac{1}{(r-e^{-i\theta})e^{i\theta}} = Ae^{-i\theta}$. Then $f(z)$ has been mapped to a point on a circle of radius $A$ encircling the origin once counter-clockwise.

However, $A$ is a function of $\theta$...

How to correctly solve this problem?

Answer 1

Suppose $\gamma$ is your contour and we parameterize it on $[a,b]$. Then we have the following.

$$2\pi n(1/(\gamma - 1; 0) = \int_{1/(\gamma - 1)} {dz/z} = \int_a^b {(1/(\gamma -1 ))'\, dt\over 1/(\gamma - 1)} = -\int_a^b {\gamma'(t)\over \gamma - 1} = -2\pi n(\gamma, 1) = 2\pi. $$

Hence $n(f\circ \gamma; 0) = 1$.