# Complex Variables

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• Mar 12th 2009, 01:53 PM
ronaldo_07
Complex Variables
http://img10.imageshack.us/img10/9859/complex2.png

Please help solve this question i do not understand it.
• Mar 12th 2009, 07:35 PM
Chris L T521
Quote:

Originally Posted by ronaldo_07
http://img10.imageshack.us/img10/9859/complex2.png

Please help solve this question i do not understand it.

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

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

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

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

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

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

Does this make sense?