Lectures on Fractional Calculus - CARMA

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The Dirac delta function• Fourier transforms integrate over extendedobjects: plane waves• Need a way of going from extended objects topoint-like objects• This is provided by the Dirac delta function:the Fourier transform of a constant2πδ(ω) = R ∞−∞ dteiωt(2π) 2 δ 2 (k x ,k y )= R ∞−∞ dxdyei(k xx+k y y)Delta function of angularfrequencyDelta function of wavevector (2D)
The Dirac delta function (2)• The more spread out a function is, the tighterits Fourier transform concentrates around theorigin• A constant is spread out uniformly in space:its Fourier transform concentrates around theorigin in reciprocal space• Another way of thinking about the deltafunction is that it is a function concentratedaround the origin, but having unit area underits curve
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 and 10: Factorial FunctionKey equations for
Page 11 and 12: A Fractional DifferentiationConundr
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: The Laplacian (2)Once:n times:∇ 2
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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