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 $x\to +\infty$.

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.