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
Best Answer
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.