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Name: benjamin heroux
Status: N/A
Age: N/A
Location: N/A
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Date: Around 1995

Derivatives and integrals are convenient to manipulate integer dimensions. However, I recently read about non-integer dimensions as they relate to fractals. Are there fractional derivatives/integrals which can be used to describe the fractals? For example,

d^n(x^p)/(dx)^n = P(p, n)x^(p-n)

Using the gamma function to find fractional factorials, you can find the coefficient which is a permutation. Therefore,

d^1.7(x^3)/(dx)^1.7 = P(3, 1.7)x^1.3

would be possible. Is there an extended math which explores fractional derivatives and integrals more thoroughly?

Yeah, although you seem to have the general idea. Another approach is through Fourier transforms. If x is the ordinary variable, and p is the transformed variable, then d/dx of a function becomes ip = pe^(i pi/2) times the transformed function. Since ip to a power is (I should say to a fractional power) is easily defined, then so is the fractional derivative. This is an approach through "analytic continuation," as is yours.


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