Find behavior and mapping forms of the following functions,
- $E(z)=e^z$.
- $T(z)=\tan z$.
Try:
- Let $z=x+iy$ and $z_0=x_0+iy_0$.
$a)$ If $x=x_0$. So $E(z)=e^{x_0}(\cos(y)+i\sin(y))$, plus $|E(z)|=e^{x_0}$ and $\text{arg} E(z)=y\ (\text{mod} 2\pi)$. This means that the function $E$ transforms vertical lines into circles centered at the origin and of radius $e^{\Re(z)}$.
$b)$ If $y=y_0$, then $E(z)=e^x(\cos y_0+i\sin y_0)$, then $|E(z)|=e^x$ and $\text {arg} E(z)=y_0\ (\text{mod} 2\pi)$. This means that the function $E$ transforms horizontal lines into rays starting from the origin.
$c)$ If $y=mx$, then $E(z)=e^x(\cos(mx)+i\sin(mx))$. Which means that the function $E$ transforms oblique lines that pass through the origin into spirals that depart from the origin with increasing radius at the rate of $e^{\Re{(z)}}$.
$d)$ If $y=y_0\ (\text{mod} 2\pi)$. So $E(z)=E(x+iy_0)$.
$e)$ Let $A=\{z=x+iy: x\in \Bbb R\ \text{and}\ y\in (-\pi,\pi]\}$. Then $E(A) =\Bbb C-\{0\}$
And I have a graph here of more or less how the mapping looks.
I would need to see the ways of mapping the $T$ that I don't know how to do, someone help me with that, please.