Given complex numbers $z_1$ and $z_2$, let $[z_1, z_2]$ denote the straight line segment path from $z_1$ to $z_2$. Recall that we can parametrize this by $x(t) = z_1 + t(z_2 - z_1)$ for $t \in [0,1]$. In the case $C=[1-i,1+i]$, sketch $C$ and evaluate the integral of the complex conjugate squared.
So I have an exam tomorrow and this topic is on it, and I have been trying so hard to figure out how to do it, but I cannot figure it out. So do I take the integral of $x(t)x'(t)$ or what. Can someone please explain to me how I do contour integration and how I would go about doing this problem. This is a big part of my exam and I really need to learn it.
Thanks so much.
Using the parametrization $\gamma\colon [0,1]\to [z_1,z_2]$ given by: $$ \gamma(t)=z_1+t(z_2-z_1), $$ we may compute the integral as follows: $$ \begin{align*} \int_{[z_1,z_2]}\overline{z^2}\ dz&=\int_{t=0}^1\overline{\left(z_1+t(z_2-z_1)\right)^2}\gamma'(t)\ dt\\ &=(z_2-z_1)\overline{\left.\frac{(z_1+t(z_2-z_1))^3}{3(z_2-z_1)}\right|_{0}^1}\\ &=(z_2-z_1)\overline{\frac{z_2^3-z_1^3}{3(z_2-z_1)}}\\ &=(z_2-z_1)\frac{\overline{z_1}^2+\overline{z_1z_2}+\overline{z_2}^2}{3}. \end{align*} $$