Representing a complex curve without software

Question

How to represent in the complex plane the curve $$\{\frac{1+i}{\sqrt2}e^{-3t-it}:t\ge0\}?$$ I did it using Maple but I don't know how to do it on paper.

Answer 1

In polar form,

$$\rho\,e^{i\theta}=e^{-3t}e^{-i(t+\pi/4)}$$

giving in polar coordinates

$$\begin{cases}\theta=-t+\frac\pi4,\\\rho=e^{-3t}\end{cases}$$

and after elimintation of $t$,

$$\rho=e^{3\theta-3\pi/4}$$

which is a logarithmic spiral.