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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?

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$E_\mu^t$ 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 $\ln_1 x= \ln x$.

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?

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 $E_{1.3333}^t \ne e^t$.

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

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?

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

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.

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