The 0ᵗʰ through 1ˢᵗ fractional derivatives [ i.e. α∈(0,1) ] are actually defined through an integral relation. (note the familiar structure to Cauchy's Integral Formula)pic.twitter.com/FwJSaaWzeL
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Visualizing the Fractional Derivatives of sin(x) • We see that the aᵗʰ derivative of sin(x) smoothly transitions between the familiar sin(x) and cos(x) when a∈[0,1]pic.twitter.com/DFBoRs8MmF
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I like your gif... it is the best motivation for this topic.
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TIL
about fractional differential operators, and tangentially this uncelebrated autodidact Oliver Heaviside to whom so much is apparently owed.
https://en.m.wikipedia.org/wiki/Fractional_calculus …
Also cool that these can be extended to work for complex a ∈ C too. -
this can also be done through Fourrier transform by convolution. at least for compact function, no ?
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so... to cos and sin the aᵗʰ derivate is the same as f(x)↦f(x+a)
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i don't think the phase change is exactly a.. maybe a*π/2 though..
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This one is called as Liouville Definition of the Fractional order integral/derivative. There's yet another called the Grunwald-Letnikov definition, which is generalized as a limit of sum. Fractional derivative of any sinusoid wave does not exist on that!
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Is there any relationship between fractional derivation and fractal analysis?
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