Rotate function in complex plane

Question

I'm pretty new to to complex numbers and did some exercises with rotating a point by certain degrees. For example if I have the point $4+2i$ I can rotate it by 20° with $(4+2i)*(\cos(20)+i*\sin(20))$, right? WolframAlpha can plot this quite nice. But how would I plot rotating an entire function?

I read here that to rotate a sine wave by 45° you do $e^{i \pi n} * \frac{1+i}{\sqrt{2}}$, or did I misread that? Because WolframAlpha doesn't show a rotation. If it's correct, where can you plot things like this? And how would you rotate by 20° or any other amount?

To go one step further with the expectation that it gets complicated, how would you create a plot like this (bottom left), where you wrap a sine wave around an axis? Not talking about the animation, just the plot at one step.

Answer 1

For rotate any function in complex plane, first you must rewrite it with complex numbers where $x$ is real part, and $f(x)$ is imaginary part. For example, for sinusoid $f(x)=\sin(x)$ we get $x+i \sin (x)$. In this step, $x$ not mean x coordinate more, now $x$ mean parameter, and good practice change it to another variable: $t+i \sin (t)$. After that we can get coordinates back as $x=\Re(t+i \sin (t))$ and $y=\Im(t+i \sin (t))$. Second, multiply your complex function with $\exp(i a)$, where $a$ is rotating angle: $(t+i \sin (t)) e^{i a}$. Now you can draw rotated function using parametric plot. With Wolfram Mathematica it loоk like this

Manipulate[
ParametricPlot[
ReIm[(t + I Sin[t])E^(I a)]
,{t,0,4 Pi}]
,{a,0,2 Pi}]

Answer 2

I think the linked question is misleading. If you interpret $e^{i\pi n}$ for $n \in \mathbb R$ the same as you interpret $4 + 2i,$ it plots as a circle (not a sine wave), because $e^{i\pi n} = \cos(\pi n) + i\sin(\pi n),$ which plots on the Argand plane at Cartesian coordinates $(\cos(\pi n), \sin(\pi n)).$

Rotation of that plot just changes where you are on the circle for each $n.$ That is, after rotating by $\alpha$ radians you get the number $e^{i\pi n} (\cos(\alpha) + i\sin(\alpha)) = \cos(\pi n + \alpha) + i\sin(\pi n + \alpha),$ which plots on the Argand plane at Cartesian coordinates $(\cos(\pi n + \alpha), \sin(\pi n + \alpha)).$ There are two functions of $n$ in there, and the plot you got in Wolfram Alpha is showing you both of them. But it's really just showing you the two Cartesian coordinates of a point traveling around the unit circle.

If you want a plot in the Argand plane that looks like a rotated sine wave, start by producing a plot in the Argand plane that looks like an unrotated sine wave. Wolfram Alpha isn't showing the Argand plot at all when you just enter some complex function of a variable, so you need another way to produce your plot.

You already have another answer that shows the correct formulation, so that's a good way to get started.