Fractals and non-integer dimensions
Name: benjamin heroux
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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