So I'm new to Complex Analysis and this question about range is driving me mad...
Find (describe) the range of the complex function:
$f(z) = {e^{2z}},\mathrm{\ for \ z \ in \ the \ rectangular \ region \ }\{z = x +iy \ | \ 0 \le x \le \ln(2), {\pi \over 4} \le y \le {\pi \over 2}\}$
It is easy enough to rewrite $f(z)$ in terms of $x$ and $y$ in standard form and then define $w = u(x,y) + iv(x,y)$.....
$f(z) = {e^{2z}} = {e^{2x}}{e^{2iy}} = {e^{2x}}\cos2y + i{e^{2x}}\sin2y$
and so
$u(x,y) = {e^{2x}}\cos2y, \ v(x,y) = {e^{2x}}\sin2y$
But I am not sure how to proceed from here :(
If you think in polar co-ordinates you can stop at ${e^{2x}}{e^{2iy}}=r{e^{i\theta}}$
The first part is the distance from the origin, and the second part is the angle. So the distance from the origin ranges from ${e^{0}=1}$ to ${e^{2\ln(2)}}=4$, and the angle ranges from $\pi\over 2$ to $\pi$. The range is the entire "block arc" shape this covers.