Complex integration confusion

Question

I wish to compute $\int_{C}(x^2 - iy^2)dz$, where $C := \{z\mid |z|=1\}$ is positively oriented.

I am a bit confused on what $dz$ actually is.

I know I have $\int_{C}x^2dz - i\int_C y^2dz$, but I still need to know what $dz$ is so I can differentiate in terms of $x,y$. Or maybe I put $x,y$ in terms of $z$.

$z = x+iy$, so then $dz = (x+iy)dx,\; (z+iy)dy$?

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So the definition of complex integration is:$$\int_{C}f(z)dz = \int_{C}(u+iv)(dx+idy)$$ $$ = \int_{C}(udx - vdy) + i\int_{C}(vdx + udy)$$ where $f(z)=x+iy$.

So how do we deal with the contour?

Answer 1

Parametrize $C$ by setting $z = e^{it}$, $0 \le t \le 2\pi$. Then $x = \cos t$, $y = \sin t$, and $$dz = dx + i\, dy = (-\sin t - i\cos t)\, dt.$$ Thus

\begin{align}\int_C (x^2 - iy^2)\, dz &= \int_0^{2\pi} (\cos^2 t - i\sin^2 t)(-\sin t - i\cos t)\, dt\\ & = \int_0^{2\pi} [-(\cos^2 t\sin t+\sin^2t \cos t) + i(\sin^3 t - \cos^3 t)]\, dt,\\ \end{align}

which I leave to you to compute.