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Intuitive essence of fractional derivatives: Interpret the n-th derivative as the coefficient of \epsilon^n in a "generalized" Taylor expansion of f(x+\epsilon)
by ssivark 11y ago
Intuitive essence of fractional derivatives: Interpret the n-th derivative as the coefficient of \epsilon^n in a "generalized" Taylor expansion of f(x+\epsilon). If the "function" behaves weirdly enough to not have a nice Taylor expansion with integer powers, then you can essentially extract the behaviour under increments by \epsilon. HTH. (IANAMathematician :-)
- mdevere 11y agoIs the author's definition consistent with that?
- magicalist 11y agoAt first look it appears to be yes, but (and maybe the GP can explain more) I don't see how it really gives any insight into the fractional derivative. It's just subbing out the derivative operator for finite differences, and you'd still need to introduce the gamma function, so you aren't gaining much that way either. What am I missing here?
- yablak 11y agoFrom the functional form of the author's definition, it looks like yes. However, to connect the two definitions probably requires using Cauchy's integral formula (http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula) and/or the complex method of residues, along with some line integral tricks (http://en.wikipedia.org/wiki/Methods_of_contour_integration http://en.wikipedia.org/wiki/Methods_of_contour_integration).
- yablak 11y agoBTW, I believe the "generalization" of the taylor expansion mentioned here is the Laurent series (http://en.wikipedia.org/wiki/Laurent_series http://en.wikipedia.org/wiki/Laurent_series).