Lectures on Fractional Calculus - CARMA

Technical details (2)We write down the Fourier transform of r β :r β = R ∞ R ∞ Γ(1+β/2)e 2πi(k xx+k y y) dk−∞ −∞ Γ(−β/2)π β+1 k β+2 x dk y .To check this expression, convert the double integral toan integral over angle multiplied by an integral over kdk.The integral over angle gives the Bessel function2πJ 0 (2πkr). You then get2Γ(1+β/2)R ∞k −1−β dkJΓ(−β/2)π β 00 (2πkr).You then use Weber’s integral to verify the result.