$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$
which is the Image of this curve and / or what is the geometrical description of the set of points
$ z : z=\gamma_1(t)$
I am helping a friend study Complex Analysis and we've come across this exercise, in the Complex / Contour Integration Chapter and we would welcome any help as we are not sure what to do.
The same is required for $\gamma_2(t)= t^3 + i\, |t^3| , t\in [ -1, 1]$ but my guess is that the methodology will be the same as above.
Please excuse my english. In my language the term used is translated in orbit or trajectory of the curve. Reading the definition given in the plane-curves section, I think that the term is curve, but I'm referring to the image of these $\gamma_1,\gamma_2$ functions.
Recall that you can identify $\mathbb C$ with $\mathbb R^2$: the point $x+iy$ corresponds to the point $(x,y)$. So the problem is to draw the image of the function $t \mapsto (t^2,t^4)$ in the plane.
You might first try finding the image of the curve for $t \in [0,1]$. Remember that the image of the curve does not change if you reparameterize the curve, so $\gamma_1([0,1])$ is the same as the image of the map defined by $s \mapsto (s,s^2)$ for $s \in [0,1]$. To get the image for $t \in [-1,0]$, think about how to use the symmetries of the defining functions.