Paramtrizing a counterclockwise circle vs. a clockwise one

Question

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane?

For example, I am looking at computing an integral $\int_\gamma {1\over{z+4}}dz$ where $\gamma$ is the circle of radius $1$, centered at $-4$, oriented counterclockwise.

My parametrization look like this: $\gamma(t)=p+Re^{it}=-4+e^{it}, 0\leq t\leq 2\pi$. Would the parametrization look the same as well if the circle oriented clockwise?

I have the final answer for integral as $2\pi i$, which makes sense, would it be the same regardless?

Answer 1

Clockwise, the paramatrization looks like

$$\gamma(t) = -4+e^{-i t} \qquad 0 \le t \lt 2 \pi$$

The integral is then

$$-i \int_0^{2 \pi} dt \, e^{-i t} \, e^{i t} = - i 2 \pi$$

Thus, changing orientation changes the sign of the integral, as you'd expect.

Answer 2

Changing orientation changes sign.