Lectures on Fractional Calculus - CARMA

A Neat TrickWe know the Green’s function for the Laplacian:G 2 (r) =− 12π ln(r)gives∇ 2 G 2 (r) =−δ 2 (x, y)We want to know what G 2s is for which∇ 2s G 2s (r) =−δ 2 (x, y)We write∇ 2 G 2 (r) =∇ 2s (∇ 2−2s G 2 (r))Then quite simply:G 2s (r) =∇ 2−2s G 2 (r)So all we need is to apply the Laplacian to an arbitrarypower to the log function!