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Extra resources for A Guided Tour of Mathematical Physics
3 The fractional derivative in terms of ﬁnite diﬀerences the Gr¨ unwald-Letnikov derivative Extending a general rule from n ∈ N to α ∈ C is a very helpful recipe to establish another deﬁnition of a fractional derivative in terms of a limit of ﬁnite diﬀerences. 50) h→0 ∞ = exp(kx) lim 1 − h→0 (−1)j j=0 ∞ = exp(kx) lim − h→0 (−1)j hj−1 j=1 = exp(kx) lim k − h h→0 (hk)j j! kj j! k3 k2 + h2 ... 2! 3! 54) which nicely coincides with the Liouville deﬁnition of a fractional derivative. 45) we have presented a ﬁrst unique deﬁnition of a fractional derivative, which is valid for any analytic function, as long as the series converges.
2 Mittag-Leﬄer functions Besides the gamma function Euler has brought to light an additional important function, the exponential: ∞ ez = zn n! 85 X α Fig. 2 Solutions and zeroes of the Caputo-wave equation are the fractional pendant of the trigonometric functions and special cases of the Mittag-Leﬄer function cos(α, x) = E2α (−x2α ) and the generalized Mittag-Leﬄer function sin(α, x) = xα E2α,1+α (−x2α ). The graph is given for diﬀerent α near α = 1. Units are given as multiples of π/2. 14) where we have introduced an arbitrary real number α > 0.
All these observed phenomena have diﬀerent physical causes. Within the framework of Newton’s theory they are summarized as friction forces FR . To be a little bit more speciﬁc we consider as friction forces all kinds of forces which point in the opposite direction of the velocity of a particle. Therefore an ansatz for friction forces is a simple power law: FR = −μ sign(v)|v|α with an arbitrarily chosen real exponent α. 8) November 8, 2013 17:18 BC: 8934 - Fractional Calculus HerrmannFC2˙main Friction Forces 35 α ≈ 0 is observed for static and kinetic friction for solids α = 1 Stokes friction in liquids with high viscosity α = 2 is a general trend for high velocities In reality both gases and liquids show a behavior which only approximately corresponds to these special cases.