Can anyone provide or give an expression in the sense of distribution theory for the functions $|x|^{s} , \log|x| $? I mean I would like to evaluate the Fourier transform $ \int_{-\infty}^{\infty}f(x)\exp(-iux) $ of these transforms in case it is possible.
[Math] Fourier transform of $ |x|^{s} $ and $\log|x| $
fourier analysisintegral-transforms
Best Answer
Concerning functions in question are not integrable on the line, the Fourier transform has to be considered in the sense of distributions. Particularly for the logarithm, it is known that (Vladimirov, Equations of Mathematical Physics, $\S2.5$) $$ F\left[{\cal P}\frac1{|x|}\right]=-2\gamma-2\log|\xi|, $$ where $\gamma$ is the Euler constant and ${\cal P}\frac1{|x|}$ is a distribution defined by $$ ({\cal P}\frac1{|x|},\varphi)= \int_{|x|\le 1}\frac{\varphi(x)-\varphi(0)}{|x|}\,dx+ \int_{|x|> 1}\frac{\varphi(x)}{|x|}\,dx. $$ With inverse FT one can get from here the FT of $\log|x|$: $$ F[\log|x|](\xi)=-2\pi\gamma\delta(\xi)-\frac\pi{|\xi|}, $$ taking into account that FT is defined in this book as $$ F[f](\xi)=\int_{\mathbb R}f(x)e^{ix\xi}\,dx. $$