$$f(\omega,u):=\frac1{\omega+iu}$$
where $i$ is the imaginary unit number. We see that the integral of a Fourier transform
$$\int\_1^\infty du\int\_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=2\pi i\int\_1^\infty du\,e^{-ux}=2\pi i\frac{e^{-x}}x$$
for $u>0$, $x>0$, while interchanging the order of integration,
$$\int\_{-\infty}^\infty d\omega \,e^{-i\omega x}\int\_1^\infty duf(\omega,u)$$
is not even integrable in the first integration. Is there a theory, say distribution or generalized function, to treat this kind of double integral involving the Fourier transform?


On the other hand, if we integrate by part the Fourier transform
$$\int\_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=\frac1 {ix}\int\_{-\infty}^\infty d\omega\,\partial\_\omega f(\omega,u)\,e^{-i\omega x},$$
$$\partial\_\omega f(\omega,u)=-\frac1{(\omega+iu)^2}.$$
We interchange the order of integration and obtain a finite value
$$\int\_1^\infty du\,\partial\_{iu} f(\omega,u)=if(\omega,u=1)=\frac i{\omega+i}$$
which is already shown to be Fourier transformable. It is easy to check the two orders of integration agree.


How do we turn this trick into a coherent theory? Does, say tempered distribution, help?



Interchanging Integration Order involving Fourier Transform
The subject of definite integrals for distributions was investigated in some detail by several mathematicians in the 50’s—-they used an elementary approach based on the fact that distributions are (locally) higher derivatives of continuous functions. This allows a treatment at the level of a (european) freshman calculus course, without using Schwartzian duality. You can find a lucid version online by googling the title „Integration and orders of growth of distributions“ of an article by J. Sebastião e Silva. The good news is that the integral you are looking at converges in the distributional sense and can be calculated as an iterated one as you ask—-this is 14.4 on p. 365.
(The article isn‘t that long—-the pagination is due to the fact that it is one of a collection).


