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How would I evaluate $\int \bar z dz$, with

1: the contour $\gamma$ being the straight line segment from $0$ to $1+i$

2: the contour $\sigma$ being the straight line segment from $0$ to $1$, followed by the straight line segment from $1$ to $1+i$.

For the first one, I tried working it from the definition $\int_\gamma f(z) dz =\int_a^bf(\gamma(t))\gamma'(t) dt$. The only question I have is, is defining $\gamma(t)=t+it$, $t\in [0,1]$ correct?

For the second one, how would I define the contour $\sigma$ and thus evaluate $\int \bar z dz$? I can't seem to be able to define $\sigma$.

Alternatively, is there any other way to evaluate $\int \bar z dz$ apart from going straight from the definition?

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up vote 3 down vote accepted

Your $\gamma$ for 1) is fine. For 2), just choose a path which connects $0$ and $1$ on, say, $[0,1]$ and $1$ and $1+i$ on $[1,2]$ such that path is continuous and piecewise differentiable on $[0,2]$.

You could, for example, define $\sigma(t) := t$ (which is the same as $ t +0i$) for $0 \le t \le 1$ and $\sigma(t) := 1+ (t-1)i$ for $1 \le t \le 2$

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Thanks @Thomas , for the path connecting $0$ and $1$, I should choose $t$? But how would I choose a path for $1$ and $1+i$. Is it $it$? – Derrick May 26 '12 at 11:34
Yes to both questions, of course with $\,0\leq t \leq 1$ – DonAntonio May 26 '12 at 11:42
added a possible parametrization of the path to the answer. – user20266 May 26 '12 at 11:43

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