The modified Bessel function of the second kind K$\nu$ appears in a wide variety of applied scientific fields. While its use is greatly facilitated by an implementation in most numerical libraries, overflow issues can be encountered especially for large value of $\nu$. After giving some necessary and sufficient conditions for their occurrences, this technical note shows that they can mostly be avoided by directly computing the logarithm of K$\nu$ thanks to a simple and stable forward recursion. A statistical examples based on the Gil-Pelaez inversion formula is given to illustrate the recursive method.

翻译：第二类修正贝塞尔函数K$\nu$广泛应用于众多应用科学领域。尽管大多数数值库中的实现极大地方便了其使用，但尤其在$\nu$值较大时，可能会遇到溢出问题。本技术说明在给出溢出发生的充要条件后，表明通过一种简单且稳定的前向递推直接计算K$\nu$的对数，通常可以避免此类问题。文中基于Gil-Pelaez反演公式给出了一个统计示例，以说明该递推方法。