Today David Udell asked me the following question:
We can have n-derivatives, defined as
dndxnf(x)=ddx⋯ddxn timesf(x)
but this definition only works for integer n, how could we extend the notion of derivative to non-integer n.
After some thinking I came up with the following...
For convenience, denote dndxn as Dn. Then if f is a degree m polynomial, this space is finite dimensional, and D is a linear map. We can define
eAt=I+At+12(At)2+⋯+1k!(At)k+⋯
where A is a matrix, and thus if eA=D, we have
logD=A
and can write
Dt=etlogD=eAt
for t∈R.
Now to evaluate the tth order derivative at a point a of an arbitrary function (which can be differentiated at least t times), we can taylor expand around a, take the derivative matrix for degree t polynomials
D=⎡⎢
⎢
⎢
⎢
⎢⎣010⋯0002⋯0⋮⋮⋮⋱⋮000⋯t−1⎤⎥
⎥
⎥
⎥
⎥⎦
then find its logarithm, and take etlogD.
I knew fractional derivatives were a thing, but not how they were defined or their use cases/properties, which was why I thought it was a fun question.
Maybe they are trying to invent an alternative definition of fractional derivatives that aren’t.
After all, one can introduce as many definitions as they like as long as those converge on integer powers.
10Kaarel
I think this approach doesn't make sense. Issues, briefly:
* if you want to be squaring D, you need it to be square — you should append another row of 0s
* this matrix D does not have a logarithm, because it isn't invertible ( https://en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence )[1]
* there in fact isn't any matrix X that could reasonably be considered a D3/2, because such an X should satisfy X2=D3, but the matrix D3 does not have a square root (see e.g. https://math.stackexchange.com/a/66156/540174 for how to think about this)
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1. also, note that it generally doesn't make sense to speak of the log of a matrix — a matrix can have (infinitely) many logarithms ( https://en.wikipedia.org/wiki/Logarithm_of_a_matrix#Example:_Logarithm_of_rotations_in_the_plane ) ↩︎