Modified Bessel function of the second kind ($K_{\nu }$)

The modified Bessel function of the second kind is the solution $K_{\nu }(x)$ of the modified Bessel differential equation that decays exponentially as $x\to +\infty$.

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.

For real $x>0$ and $\nu$, one useful representation isIt turns large-order asymptotics into a Laplace integral with a moving maximum.

For fixed $x>0$,as $\nu \to +\infty$. Applying Laplace's method to the integral representation of the modified Bessel function of the second kind proves the formula.