Parametrizing this curve

Question

How can I parametrize the trajectory

so that it is a smooth path $h:[-1,1]\rightarrow \mathbb{C}$?

I think that I should use $$h=\left\{\begin{array}{ccl}t+i |t|&:&-1\leq t \leq 0\\ ? &:&0\leq t \leq 1\end{array}\right.$$ but I'm not sure what to put for $?$ to make it smooth.

I apologize in advance if this is a duplicate. I couldn't find it in the search.

Answer 1

The parametrization $h : t \mapsto \left\{ \begin{matrix} \exp \left( 1- \frac{1}{t^2} \right) (1,-1) & \text{if} \ t<0 \\ \exp \left( 1- \frac{1}{t^2} \right) (1,1) & \text{if} \ t>0 \\ 0 & \text{if} \ t=0 \end{matrix} \right.$ is even $\mathcal{C}^{\infty}$. However, the derivative of any parametrization is zero at $(0,0)$ since the slope of the tangents is discontinuous at this point.

Answer 2

Your path is not smooth. Under no parametrization will it become smooth.