Order of integration of a Fourier transform of $ 2 $ D

Suppose that the $ 2 $D The Fourier transform of a PDE can be written as, $$ hat {u} (a, b) = – frac {1} {2 pi} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} frac {f ( alpha, beta)} {a ^ 2 + b ^ 2} e ^ {- i (a alpha + b beta)} d alpha d beta. $$ Now, it's the inverse Fourier transform of $ hat {u} (a, b) $ equivalent to $$ u (x, y) = – frac {1} {4 pi ^ 2} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} frac {f ( alpha, beta)} {a ^ 2 + b ^ 2} e ^ {- i (a? + b?) + i (ax + by)} d alpha d beta da db $$ or $$ u (x, y) = – frac {1} {4 pi ^ 2} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} int _ {- infty} ^ { infty} frac {f ( alpha, beta)} {a ^ 2 + b ^ 2} e ^ {- i (a? + b?) + i (ax + by)} gives db d alpha d beta. $$

I am not sure of the correct order of integration.