I´m stuck trying to find the image of the circle $|z|\leq\frac{\pi}{2}$ under the function $f(z)=e^z$. I know the transformation results in $w=f(z)=u+iv$, with $u = e^x\cos x$ and $v=e^x\sin x$, having both $x$ and $y$ restricted in the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.
But it has to look like this. I don´t know how to get a parametric solution for it. Thanks in advance.
EDIT: This is a problem from Complex Analysis by Gamelin, and it appears in a section prior to the introduction of differentiation, so conformal mapping (wich I understand is an application of complex differentiation) is not an acceptable solution. It must be a way to obtain the same equation by using classical mapping methods
Continuously differentiable (in particular, holomorphic) applications map boundaries into boundaries, hence it is enough to understand the image of $|z|=\frac{\pi}{2}$, made by points of the form $\frac{\pi}{2}e^{i\theta} = \frac{\pi}{2}\cos\theta + i\frac{\pi}{2}\sin\theta $ for some $\theta\in[0,2\pi)$. By exponentiating this thing we get
$$ e^{\frac{\pi}{2}\cos\theta}\left[\cos\left(\frac{\pi}{2}\sin\theta\right)+i\sin\left(\frac{\pi}{2}\sin\theta\right)\right] $$ i.e. a regular curve contained in the annulus $e^{-\pi/2}\leq |z|\leq e^{\pi/2}$ and in the half-plane $\text{Re}(z)\geq 0$:
$\hspace1in$
It is a sort of cardioid.