We have the following Fourier transforms: $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{1/2}}\right] = 1/\sqrt{k_x^2+k_y^2} $$ $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{3/2}}\right] = -\sqrt{k_x^2+k_y^2} $$ $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{5/2}}\right] = \frac{1}{9}\sqrt{(k_x^2+k_y^2)^3} $$
I am interested in obtaining $$ {\cal F}\left[\frac{x}{(x^2+y^2)^{1/2}}\right],\quad {\cal F}\left[\frac{x}{(x^2+y^2)^{3/2}}\right],\quad {\cal F}\left[\frac{x}{(x^2+y^2)^{5/2}}\right] $$
I would appreciate your suggestion/insight.
HINT for the last two:
$$ \frac{x}{(x^2 + y^2)^{3/2}} = - \frac{d}{dx} \left[ \frac{1}{(x^2 + y^2)^{1/2}} \right] $$ $$ \frac{x}{(x^2 + y^2)^{5/2}} = - \frac{1}{3} \frac{d}{dx} \left[ \frac{1}{(x^2 + y^2)^{3/2}} \right] $$ How is the Fourier transform of a function related to the Fourier transform of its derivative?