In the number theory we have roots of unity, i.e. the $n$ solutions of:
$$x^n=1,~~~n=1,2,3,\dots$$
$$x_1=1$$
$$x_2=(-1,1)$$
$$x_3=\left( 1, e^{\dfrac{\pi i}{3}}, e^{\dfrac{2 \pi i}{3}} \right)$$
$$x_4=(1,-1,i,-i)$$
etc.
If we consider a simple ODE:
$$f^{(n)}(x)=f(x),~~~n=1,2,3,\dots$$
We obtain a something very similar (in my opinion):
$$f_1(x)=C e^x$$
$$f_2(x)=C_1 e^x+C_2 e^{-x}$$
$$f_3(x)=C_1 e^x+C_2 e^{-x/2} \sin \left( \frac{\sqrt{3}}{2} x \right)+C_3 e^{-x/2} \cos \left( \frac{\sqrt{3}}{2} x \right)$$
$$f_4(x)=C_1 e^x+C_2 e^{-x}+C_3 e^{ix}+C_4 e^{-ix}$$
etc.
We can always switch between the exponential and trigonometric forms of course.
Do these functions play as important role in functional analysis or other fields, as the roots of unity play in number theory? What are they called? What are some examples of their use?
(I mean the whole set of these functions, not just the exponent or trig functions).
If possible, please offer an intuitive explanation of the connection/analogy between roots of unity and these functions.
When we think of the differential equation $y^{(n)} = y$ we immediately think of the exponential function $e^x$. Let us generalize a bit and consider instead $e^{a\cdot x}$ for some $a\in \mathbb{C}$.
If we derive $n$ times this function we will get $a^n e^{a\cdot x}$ therefore in order for this function to satisfy the differential equation, we need to have $a^n = 1$.
Note also that if we can find $n$ different values $a_1, a_2, ... , a_n$ such that all $e^{a_i\cdot x}$ satisfy the equation we win because all these function are linearly independent.