What would this contour look like in the complex plane?

Question

I'm currently trying to visualise what the contour $\gamma (t) = te^{4 \pi i t}$ (for $0 \leq t \leq 1$) would look like in the complex plane. I would imagine that it would be a spiral, twice anti-clockwise starting at the origin, but I am not sure.

Is my intuition correct? If not, what should this look like and how can I better understand/visualise contours like this before attempting to sketch them?

Answer 1

Yes it is a spiral starting in the origin, winding around it exactly two times. You could plot the parametric version for a better visualization:

$$\vec x(t)=\begin{pmatrix}t \cos(4\pi t) \\ t \sin(4\pi t)\end{pmatrix}.$$

Answer 2

Complementing @M.Winter's answer, here is a graphic of the parametric equations: $$\left\{ \begin{aligned} x(t) &= t \cos(4\pi t) \\ y(t) &= t \sin(4\pi t) \end{aligned} \right. $$

An Archimedean Spiral.