CVXI2

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Compute \int _{\Gamma }{\mbox{Re}}\ z\ dz along the directed line segment from z=0\, to z=1+2i\,.

Parametrize the directed line segment by z(t)=t(1+2i),0\leq t\leq 1. Then

\int _{\gamma }f(z)\ dz =\int _{a}^{b}f(z(t))z^{\prime }(t)\ dt
\int _{\Gamma }{\mbox{Re}}\ z\ dz =\int _{0}^{1}{\mbox{Re}}(t+2ti)\cdot (1+2i)\ dt
=(1+2i)\int _{0}^{1}t\ dt
={\frac  {1+2i}{2}}t^{2}\left.\right|_{0}^{1}
={\frac  {1+2i}{2}}
={\frac  {1}{2}}+i


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