Fourier transform (logarithm) question Can we think, at least in the sense of distribution, about the Fourier transform of $\log(s+x^{2})$? Here '$s$' is a real and positive parameter
However $\int_{-\infty}^{\infty}dx\log(s+x^{2})\exp(iux)$ is not well defined.
Can the Fourier transform of logarithm be evaluated ??

The answer is given here; $a$ is assumed to be positive, so $a=\sqrt s$:
$$\mathcal F\!\left[ \ln\left( x^2 + s \right) \right] =
-\sqrt{2 \pi} \left( \left| w \right|^{-1} e^{-\sqrt s \left| w \right|} +
 2 \gamma \delta(w) \right).$$
Also, with $\left| x \right|^{-1}$ defined in the same way, for negative $s$ we have
$$\mathcal F\!\left[ \ln\left( x^2 + s \right) \right] =
-\sqrt{2 \pi} \left( \left| w \right|^{-1} \cos \left( \sqrt {-s} w \right) +
 2 \gamma \delta(w) \right).$$
