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1. Feb 6, 2015

bubblewrap

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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2. Feb 6, 2015

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 $\ln_1 x= \ln x$.

3. Feb 7, 2015

bubblewrap

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?

4. Feb 10, 2015

the_wolfman

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

5. Feb 11, 2015

bubblewrap

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

6. Feb 11, 2015

the_wolfman

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?

7. Feb 12, 2015

bubblewrap

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

8. Feb 15, 2015

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.

9. Feb 23, 2015

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