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

The Modified Bessel function of the second kind $K_{\nu }(x)$ is the standard solution of the Modified Bessel differential equation normalized, for fixed real order and $x\to +\infty$, bySaying only that it decays would leave an arbitrary multiplicative constant; the displayed coefficient fixes the convention. Its integral representation of the modified Bessel function of the second kind giveswhich is positive for real $\nu$ and $x>0$.^[ref]

For order one half, the exact expression is $K_{1/2}(x)=\sqrt{\pi /(2x)}{e}^{-x}$. At order zero, $K_0(x)$ instead has a logarithmic singularity as $x\to 0+$ despite decaying at infinity. Thus a decay condition at one endpoint does not imply regularity at the other.

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.