system
Given the following post, respond as if you're commenting on Reddit:
human
f(x) = e^x, f(x) = sin(x) and f(x) = 0 are functions which arrive at itself again when derived (multiple times). Are there more such functions? Is this a relevant class of functions?
gpt
You are looking for functions that satisfy the differential equation

> f^([n]) = f

for some positive integer *n*. For each *n*, this is an *n*th-order, linear differential equation. So the solutions form an *n*-dimensional vector space. In particular, there are only finitely many linearly independent such functions for each *n*.

The equation has constant coefficients, and so it can be easily solved. If ω*_n_* is a primitive *n*th root of unity, then the linearly independent solutions are

> f*_k_*(x) = exp(ω*_n_*^(k)x)

for k = 0, 2, 3, ..., *n*-1.

For n = 1, we get just e^(x). For n = 2, we get e^(x) and e^(-x). For n = 3, we get, after taking suitable linear combinations,

> e^(x)

> e^(-x/2)cos(√3/2 x)

> e^(-x/2)sin(√3/2 x)

For n = 4, we get e^(x), e^(ix), e^(-x), and e^(-ix). So after taking suitable linear combinations, we can get sin(x) and cos(x). If you are wondering where 0 fits into this, remember that, for any *n*, we can take linear combinations of such functions to get another such function. The zero function is a linear combination of *any* function.

You can then take it from there.

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Also, as a side note, the English word meaning "to take the derivative of" is "to differentiate". So we don't say functions which arrive at itself again when derived", we say ".... when *differentiated*". While it's true that the nominative and verbal forms of "derivative" have the same root in many other languages, they don't in English for some reason.