Suat AyözProblem for May 8th from the 2026 AMS Daily Epsilon of Math Calendar
Problem for May 7th from the 2026 AMS Daily Epsilon of Math Calendar
StylusFalling asleep the other night I was thinking about derivatives.
I can think of two functions that are their own derivatives: f(x) = 0 and f(x) = ex.
There's a pair of functions function such that each is its second derivative is itself: f(x) = e-x and g(x) = -e-x.
There's a group of functions such that its fourth derivative is itself, f(x) = sin(x), g(x) = cos(x), h(x) = -sin(x), k(x) = -cos(x)
Are there any functions with other numbers of derivative steps before they become equal to themselves again? 3, 5, 8, etc.
In the light of day, I think I see that we could define an "n-self-derivative function" by just setting the derivatives
f(0) = 1
f'(0) = -1/2
f''(0) = -1/2
f'''(x) = f(x) etc
and that defines a unique Maclurin series.
But .. are there functions of this form that would arise naturally?
Update: Yes, one such function is exp(-x/2)*cos(x*sqrt(3)/2). see the response from @Chip_Unicorn: https://im-in.space/@Chip_Unicorn/116529978344298695
Chip Unicorn (@[email protected])
@[email protected] Math stack exchange has the answer: https://math.stackexchange.com/questions/1080016/function-whose-third-derivative-is-itself The answers are interesting!
Problem for May 6th from the 2026 AMS Daily Epsilon of Math Calendar
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Problem for April 30th from the 2026 AMS Daily Epsilon of Math Calendar
