Understanding Euler's Identity (complex) For a complex number $z=a+bi$ and a positive real value $R$, we have $e^{Rbi}=\cos(Rb)+i\sin(Rb)$. I am struggling to understand this since no matter how large $b$ or $R$ is, we have $|e^{Rbi}| \in [-1, 1]$. What is the best way to understand this intuitively? For instance, in an applied sense, is it true that
$$\bigg|\sum_{z: \ a, b \geq 0}e^{Rbi}f(z)\bigg|\leq \bigg|\sum_{z: \ a, b \geq 0}f(z)\bigg|,$$
where $f$ is some generic function and $\sum_{z: \ a, b \geq 0}f(z)>0$? This makes sense to me because each $|e^{Rbi}|$ is no larger than $1$.
 A: Starting with the basics, $t\mapsto e^{it} = \cos t +i\sin t$ just describes the complex unit circle, which for $t\in\mathbb{R}$ is run through counter-clockwise with constant speed $1$. This just means that one cycle around the circle takes $2\pi$ time units, which is the length of the unit circle.
So what happens if we plug in $2$, i.e. $e^{i2t}$? Now the speed is $2$, which means one cycle takes $\pi$ time units.
So by increasing the factor $R\in\mathbb{R}$, the geometric shape of the curve traced out by $t\mapsto e^{iRt}$ is always the unit circle. The only thing that changes is the speed with which this circle is orbited.
This means that $|e^{Rit}| = $ for all $t,R\in\mathbb{R}$, since we never are able to leave the circle.
A: Based on what you have given, $Rb$ is a real number. I'll call it $\theta$. I may be interpreting the question incorrectly, but then all that is happening is the fact that $e^{i\theta}$ represents a number on the complex unit circle by Euler's formula.
The fact that $Rb = \theta$ can be arbitrary large is just the fact that any point on the circle can be specified by an arbitrary large angle $\theta$.
For example, $z=1$ can be reached by choosing an angle from the real axis $\theta = 0, 2\pi, 4\pi, 6 \pi, ..., 1000 \pi, etc.$. One could also choose the negatives of this set.
It's a matter of geometry and the periodicity of trigonometric functions that $\cos(\theta) + i \sin(\theta)$ will be confined to the unit circle.
