Parametrization of a Complex Path/Contour Integration

Question

How would I parametrize the path which is a straight line from 1 to a complex point z? Does $\delta (t) = z^t$ make any sense?

Answer 1

If $z_0$ and $z_1$ are points in the complex plane, then $\delta(t)=(1-t)z_0 + t \, z_1$ is called a convex combination or, sometimes, a weighted average of the points. When $t=0$, $\delta(0)=z_0$ and when $t=1$, $\delta(1)=z_1$. For $0<t<1$, $\delta(t)$ lies on the line segment between $z_0$ and $z_1$ - that's rather the point of the weighted average business. As $t$ ranges from $0$ to $1$, the point $\delta(t)$ moves from $z_0$ to $z_1$ tracing out that segment.

If $z_0=1$ and $z_1 = -1+i$, we get $\delta(t)=(1-t)+t\,(-1+i)$. The corresponding motion looks like so: