I have problem with parameterization of a curve in order to evaluate a complex integral. Most docs that I've tried to read didn't explain the topic very well, especially, in case where the curve in question, $C$, is a line segment from the point $z_0$ to $z_1$. For example, in a question, the curve $C$ is the line segment with initial point $-1$ and final point $i$. The author uses $z=-1+(1+i)t$ as a parameterization, but the author does not mention that how S(he) obtained the formula. In another example, the curve in question is the line segment from $0$ to $1 + i$, the author uses $z=x+ix$, but again, says nothing about how this formula is obtained. Can someone please explain the general way of parametrization of a line segment in the complex plane? Is there a book that have explained the topic in details? Also, is there something like "the equation of straight line" in the complex plane?
2 Answers
Hint:
the complex field is also a plane and complex numbers correspond to points in that plane (naturaly), i.e a $z=a+bi$ corresponds to point with $(a,b)$ coordinates in the complex plane. Any point $p$ on the line segment from point $p_0$ to point $p_1$ is parametrised as $p=(1-t)p_0+tp_1$, $t \in [0,1]$
When $t=0$, $p=p_0$, when $t=1$, $p=p_1$ and when $t \in (0,1)$, $p$ traverses all points in-between
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As for a straight line segment, the parametrization is easy: if your segment is $[x,y]$, $x,\,y \in \mathbb C$, then $$ z(t) = (1-t)x + t\cdot y = x + t(y-x), \quad t\in[0,1] $$ yields a segment with $z(0) = x$, $z(1)=y$.
Then indeed, for $[x,y] = [-1,i]$ the parametrization is $z(t) = -1 + (i - (-1))t$, and for $[0,1+i]$, $z(t) = (1+i)t$.
As for an arbitrary curve, there's no general recipe for parametrization, let alone for a "nice", compact formula, although we know formulas for many different curves. This question, somewhat tackles the topic, and this answer shows, that you can write a formula for a gorilla, but it's (usually) no easy task. Of course, as Nikos M. noted, complex parametrization is easily interchangeable with parametrization in $XY$-plane ($\mathbb R^2$).
