1. ## Complex Variables

2. Originally Posted by ronaldo_07

a) The equivalent parametric equation for the line from z=0 to z=1 is $z\!\left(t\right)=t;~0\leqslant t\leqslant 1$. Thus, $z^{\prime}\!\left(t\right)=1$.

Therefore, $\int_{\gamma}\left(z-i\right)^2\,dz=\int_0^1 t^2-2it-1\,dt$. Can you continue from here?

b) The equation of the line segment is $z\!\left(t\right)=1+it;~0\leqslant t\leqslant1$. Thus, $z^{\prime}\!\left(t\right)=i$.

Therefore, $\int_{\gamma}\left(z-i\right)^2\,dz=i\int_0^1\left(1+it-i\right)^2\,dt$. Can you continue from here?

c) The equation of the circle in parametric form is $z\!\left(t\right)=i+e^{it};~-\frac{\pi}{2}\leqslant t\leqslant0$. Thus, $z^{\prime}\!\left(t\right)=ie^{it}$.

Therefore, $\int_{\gamma}\left(z-i\right)^2\,dz=i\int_{-\frac{\pi}{2}}^0e^{3it}\,dt$. Can you continue from here?

Does this make sense?