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Evaluate

$\int_{L}\left( \overline {z}+1\right) dz$

Where L is the line segment from -i to 1+i.

On our complex analysis course we have been shown how to evaluate integral using the the FTC and using path integrals. I am struggling to apply either case here. Can someone show me the steps to find the integral.

Thanks

Tom

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Parametrize the path: $$z(t) = x(t) + iy(t) = -i(1-t) + (1+i)t,\qquad t\in[0,1]$$. $$\int_{L}\left( \overline {z}+1\right)dz = \int_0^1(\overline{z(t)}+1)z'(t)\,dt =\cdots$$

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You should parametrize your line segment as $$\gamma : t\mapsto t(1+i)+(1-t)(-i)$$ when $t\in [0;1]$. Then, you have to replace in your integral $z$ by $\gamma(t)$, $dz$ by $\gamma'(t)dt$ and $L$ by $[0;1]$ (the same as a standard change of variables for real integrals) and you should be able to compute the integral without trouble.

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