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QUESTION


Find a parametrization of the contour that follows the real axis
from 0 to 1 and then follows a straight line from 1 to $2+i$.



ANSWER


For $0\le t\le 1$, we put $z=t$. A parametrization of the line
from 1 to $2+i$ based on $[0,1]$ is $(1-s).1+(2+i)s$ To get a
parametrization based on $[1,2]$ we put $t=s+1$ or $s=t-1$.  Then
we get $2-t+(2+i)(t-1)=t+i(t-1)$. Thus the desired parametrization
is

$$\cases{z=t & $0\le t \le 1$\cr z=t+i(t-1)& $1\le t\le 2$.\cr}$$




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