Is there some technique to plot complex functions by hand?

Question

Is there some technique to plot complex functions by hand?

Particularly, consider e.g.

$$x=\exp(t)\cos(t), y=\exp(t)\sin(t)$$

This is equivalent to $e^x(\cos(y)+i\sin(y))$ along line $y=x$, just parametrized.

WA gives:

https://www.wolframalpha.com/input/?i=x%3D(exp(t))*cos(t),+y%3Dexp(t)*sin(t)

Answer 1

In your case it's relatively easy. The (unit speed) unit circle is given by $$c(t)=\begin{pmatrix}\cos(t)\\\sin(t)\end{pmatrix}.$$ The curve in question is of the form $r(t)\cdot c(t)$ where $r(t)=e^t$. That changes the radius of the unit circle so we'll get a sort of spiral.