Oh, I have two math questions that I thought about aaages ago. I don't know the answer to them and they might be interesting.
1) Think about the central limit theorem - the normal distribution you get from summing infinitely many uniform distributions.
Now think about what kind of distribution you get from summing finitely many uniform distributions. For n = 2 you get a triangular shape, for n = 3 you get a shape that looks like parabolic sections stuck together and so on.
How do you generate the formulas that create these curves for each value of n? (For normalization purposes, let's say that the domain is x >= 0, x <= 1 and the integral of the curve should be equal to 1.)
2) We know that sin and cos (and all combinations thereof), if differentiated or integrated four times, equal themselves. And we know that e^x, if differentiated or integrated once, equals themselves.
Are there any functions that equal themselves after being differentiated or integrated 2, 3, 5, 6, etc times? If so, construct them. If not, prove it's impossible.
We also know that it's possible to differentiate/integrate something a non-integer number of times. (
https://en.wikipedia.org/wiki/Fractional_calculus ) So are there any functions that equal themselves after being differentiated/integrated a non-integer number of times?
