Toronto Math Forum
MAT3342018F => MAT334Tests => Term Test 1 => Topic started by: Victor Ivrii on October 19, 2018, 04:09:49 AM

$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Find any region that is mapped bijectively to $\{w\colon \Re w\ge \Im w , \ 0< w\le 2\}$ by the map $w=e^{iz}$. Draw both of them.

The domain region is the region $\Im z \geq \ln 2, \Re z \in [\frac{3\pi}{4}, \frac{3\pi}{4}]$, or equivalently $y \leq \ln 2, x \leq \frac{3\pi}{4}$.
The range region is contained within the circle $w \leq 2$. The $\Re w \geq \Im w$ can be viewed as $x \geq y$. That means, it is the region to the right of line $x \geq y$ for the top half, and $x \geq +y$ for the bottom half.
The range region must have an absolute value of Arg less or equal to $\frac{3\pi}{4}$, and a magnitude less or equal to $2$.
It can be described as $\{w  \frac{3\pi}{4} \leq \operatorname{Arg}w \leq \frac{3\pi}{4} \wedge 0 < z \leq 2\}$
The function $e^{iz} = e^{i(x + iy)} = e^{y + ix}$. Hence, the xcoordinate determines the Argument, and the ycoordinate determines the magnitude (increases as y moves downwards).
Hence, the function $e^{iz} = (e^{y})(\cos x + i \sin x)$.
Prove that this function is injective
For $\Im z \geq \ln 2$, it follows $y \leq \ln 2$, and so the magnitude $e^{y} \leq 2$.
For $\frac{3\pi}{4} \leq \Re x \leq \frac{3\pi}{4}$, the argument of $w$ will be in $[\frac{3\pi}{4}, \frac{3\pi}{4}]$
Two nonzero complex numbers are the same when both the magnitude and principal Argument are the same. The function $e^{iz}$ transforms the real and imaginary parts of $z$ into the argument and the reciprocal magnitude of the output. If the outputs are the same (Arg, magnitude), the inputs are the same ($x$, $y$). Hence, this function is injective.
Prove that this function is surjective
Given any complex number with $0 < z \leq 2$, and $ \frac{3\pi}{4} \leq \operatorname{Arg}w \leq \frac{3\pi}{4}$. The input $x + iy$ would have the imaginary component (encodes the reciprocal of the magnitude) to be $e^y \leq \ln 2$, so $y \leq \ln 2$, and finally $y \geq \ln 2$, and the real component (encodes the Argument) to be in $ \frac{3\pi}{4} \leq x \leq \frac{3\pi}{4}$