Integrate $\int_{|z|=r} x \, dz$ I'm looking to integrate
$$\int_{|z|=r} x \, dz$$
My question is:
Given that $z(t)=re^{it}$ is a parametrization for $|z|=r$, then what's $x$? It's not a variable, because $z$ is, but does it refer to the $\cos(t)$ part of $e^{it}$ or is it a constant in the integration?

If this is given as it is, I think this is just
$$\left.x\int_{|z|=r}dz= x\int_0^{2\pi}rie^{it}dt=rxe^{it}\right|_0^{2\pi}=0$$
If there's given some definite relation between $\;x\;$ and $\;z\;$ then something else can happen say $\;x=\text{Re}\,z\;$ , and then
$$\int_{|z|=r}xdz=\int_0^{2\pi} r\cos t \,r\,i\,e^{it} dt=r^2i\int_0^{2\pi}\cos te^{it}\,dr=r^2i\int_0^{2\pi}\left(\cos^2t+i\cos t\sin t\right)dt=$$
$$=r^2i\left(\frac{t+\cos t\sin t}2-\frac14\cos2t\right)_0^{2\pi}=r^2i\pi$$
but I'm a little not sure about this last part.
