# 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.