I'm having a very hard time understanding the concept of images and mappings in the complex plane.
Considering the map $w=e^{z}=e^{x}e^{iy}$, find the image of the region $\left\lbrace x+iy:x\geq 0, 0\leq y \leq\pi \right\rbrace$. Based on my current understanding, I have rewritten $w=e^z$ by breaking it apart with Euler's Formula: $w=e^{x}\left(\cos{y}+i\sin{y}\right)=e^{x}\cos{y}+ie^{x}\sin{y}.$ From here, we know that $u(x,y)=e^{x}\cos{y}$ and $v(x,y)=e^x\sin{y}$. Could I then rewrite the mapping as $f(x,y)=\left(e^{x}\cos{y},e^x\sin{y}\right)$ in order to sketch the seperate $xy$ and $uv$ planes?