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Fractals and non-integer dimensions
Name: benjamin heroux
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
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?
Replies:
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.
jlu
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