= Fourier transform of the modified Bessel function K0
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{title2=$\widehat{K_0(|x|)}(k)=\pi/\sqrt{1+k^2}$}
The <integral representation of the modified Bessel function of the second kind> gives $K_0(x)=\int_0^\infty e^{-x\cosh t}\,dt$ for $x>0$. Taking the <Fourier transform> of the even continuation and setting $u=\sinh t$ gives $2\int_0^\infty(1+k^2+u^2)^{-1}du=\pi/\sqrt{1+k^2}$. The large-<wavenumber> terms $\pi/|k|-\pi/(2|k|^3)$ are also obtained from the local <logarithmic singularity> $-\log|x|-(x^2/4)\log|x|$ using the <Fourier transform of the absolute logarithm>.
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