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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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