\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION


Evaluate $\int_{\alpha}\bar z dz$ and $\int_{\beta}\bar z dz$ \
where $\alpha$ is the contour defined by $z=e^{\pi it}$ for $0\leq
t\leq \frac{1}{4}$ and $\beta$ is the contour define by
$z=\left\{\begin{array}{rr}t+it,&(0\leq t\leq1)\\ t+i,&(1\leq
t\leq2)\end{array}\right.$


ANSWER


$\int_{\alpha}\bar z dz=\int_0^{1/4}e^{-\pi it}.\pi ie^{\pi it}
dt=\pi it|_0^{1\over 4}=i\pi/4.$

  $\int_{\beta} \bar z
dz=\int_0^1(t-it)(1+i)dt+\int_1^2(t-i)dt={5\over 2}-i$




\end{document}