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December 4th, 2016, 03:09 PM  #1 
Newbie Joined: Dec 2016 From: England Posts: 7 Thanks: 0  Complex Integration
I am very stuck on this! i ) i worked out the straight line is y=(x1) does this help me, bit confused with the mod z when applying the integral. ii) similar follows Many thanks 
December 4th, 2016, 03:39 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,982 Thanks: 1027 
the first parameterization is $z = 1 + (1+i)t,~0\leq t \leq 1$ $z = (1+t)+it$ $z^2 = (1+t)^2 + t^2 = 2t^2 + 2t + 1$ $\displaystyle{\int_0^1}~2t^2 + 2t + 1 = \left. \dfrac{2 t^3}{3}+t^2+t\right_0^1 =\dfrac 8 3$ for the 2nd parameterization $z = (1+t) + t^2 i,~0\leq t \leq 1$ See if you can figure out the form of $z^2$ and do the integration. Note this also goes from $1 \to (2+i)$ but by a different path. 

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complex, integration, plane, smooth 
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