- #1

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- #1

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- #2

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

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

- #3

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- #4

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No. The function [itex] e^t[/itex] is always [itex] e^t [/itex], its definition does not change. The function [itex] E_\mu^t[/itex] 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).

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

- #5

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

- #6

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

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

- #7

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- #8

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The littérature on the subject is extensive. A short bibliography is given p.12 in the paper for general public :

- #9

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The random walk's guide to anomalous diusion: a fractional dynamics approach

http://www.tau.ac.il/~klafter1/258.pdf

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