Parametric equations in complex analysis

Question

I am trying to find $\int_C (1+i-2z')dz$ where$z'$ is the conjugate of $z$ and where C is the parabola $y=x^2$ from $z_1=0$ to $z_2=1+i$.

How do I write the parametric equations for this?

Answer 1

Well, you already wrote the parametrics...almost:

$$\begin{cases}x=t\\{}\\y=t^2\end{cases}\;\;\;\;,\;\;0\le t\le 1$$

and thus

$$1+i-2\overline z=1+i-2(x-iy)\longrightarrow1+i-2(t-it^2)=(1-2t)+(1+2t^2)i$$

and the complex line integral becomes

$$\int\limits_0^1\left[(1-2t)+(1+2t^2)i\right]dt\;\ldots\;\text{complete}$$

Answer 2

With $x=t$, you get $y=t^2$, and $z = x+iy = t+it^2$. You want to let $0 \le t \le 1$ to get the full curve in question.