A Question about fractional calculus

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Main Question or Discussion Point

Okay, maybe not really fractional calculus but I don't know what this stands for. Its in the black circle (more like an ellipse though), what does the mu under the natural logarithm mean?

 

[the_wolfman]

$E_\mu^t$ is a function similar to $e^t$ in that $D_t^\mu E_\mu^t= E_\mu^t$

Its often useful to define the inverse function. $E_\mu^t$ is similar to $e^t$ so we suppose that the inverse of $E_\mu^t$ should be similar to the inverse of $e^t$.

We call this new inverse function the generalized natural log, $\ln_\mu$, to highlight this similarity. In the case where $\mu=1$, $E_1^t=e^t$ and $\ln_1 x= \ln x$.

 

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What happens when$\mu$ is not 1? I don't understand how this is used. For example when $\mu$ is 2, does it mean that you use $e$ defined in $E^t_\mu$? What does that mean though?

 

[the_wolfman]

What happens when$\mu$ is not 1? I don't understand how this is used. For example when $\mu$ is 2, does it mean that you use $e$ defined in $E^t_\mu$? What does that mean though?

No. The function $e^t$ is always $e^t$, its definition does not change. The function $E_\mu^t$ is a different function, it sounds like it can be calculated using Equation 28 in the reference you posted (you didn't post the page that contains this equaion).

$E_\mu^t$ has the property $E_1^t=e^t$, but this equaility does not hold for all $\mu$. For instance $E_{1.3333}^t \ne e^t$.

 

[bubblewrap]

Yeah but as I said what happens when mu is not 1, other than it not being $e^t$?

 

[the_wolfman]

Yeah but as I said what happens when mu is not 1, other than it not being $e^t$?

When $\mu \ne 1$ the function $E_\mu^t$ is defined by equation 28 in the reference you are using. The function $\ln_\mu x$ is its inverse. What part of this is unclear?

 

[bubblewrap]

I think I understand now. One more thing, which area of physics uses fractional calculus? Is fractional calculus important in physics?

 

[JJacquelin]

There are many examples of use in physics which are reported in the book of K.B.OLdham & J.Spanier "The fractional Calculus" AcademicPress, N.-Y.

The littérature on the subject is extensive. A short bibliography is given p.12 in the paper for general public : https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire . Also, this paper shows an use of fractional differ-integration to generalize the basic electrical components.

 

[Strum]

Ralf Metzler has also worked a lot on these things. Here is a fairly straight forward introduction:
The random walk's guide to anomalous diusion: a fractional dynamics approach
http://www.tau.ac.il/~klafter1/258.pdf