Lectures on Fractional Calculus - CARMA

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Fractional Differentiation (3)d n x mdx n = Γ(m+1)Γ(m−n+1) xm−n¡d p d¢ n x mdx p dx =Γ(m−n+1)n Γ(m−n−p+1)Γ(m+1)Γ(m−n+1) xm−n−pi.e.¡d p d¢ n x mdx p dx =Γ(m+1)n Γ(m−n−p+1) xm−n−pExample:³d 1/2dx 1/2d 1/2 1dx 1/2 ´=Γ(1)Γ(1−1/2−1/2) x−1 =0
A Fractional DifferentiationConundrumWeknowforanyintegern:D n (e ax )=a n e axso we want for any αD α (e ax )=a α e ax .Yet:D α e x = D α ( P ∞n=0 xnn! )=P ∞n=0x n−αΓ(n−α+1)These don’t match unless α is an integer!
Page 1 and 2: Three Lectures on
Page 3 and 4: References• A child’s garden of
Page 5 and 6: Motivation, History (2)• Initial
Page 7 and 8: Fractional Differentiation (1)d n x
Page 9: Factorial FunctionKey equations for
Page 13 and 14: Integration as NegativeDifferentiat
Page 15 and 16: Differentiation as Negative Integra
Page 17 and 18: Adding up Harmonics• Fourier seri
Page 19 and 20: Momentum space• A lot of physics
Page 21 and 22: The Laplacian (2)Once:n times:∇ 2
Page 23 and 24: The Dirac delta function (2)• The
Page 25 and 26: Green’s functions• A Green’s
Page 27 and 28: The Green’s function in 2DWe star
Page 29 and 30: R ∞0The Green’s function in 2D
Page 31 and 32: A Neat TrickWe know the Green’s f
Page 33: Technical details (2)We write down
Page 36: • So we have deduced:G 2s (r) =

Lectures on Fractional Calculus - CARMA

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