For fixed real $\nu$ as $z\to+\infty$, the <Modified Bessel function of the first kind> and <Modified Bessel function of the second kind> have <asymptotic expansions>
$$
I_\nu(z)\sim\frac{e^z}{\sqrt{2\pi z}}\sum_{j=0}^\infty\frac{(-1)^ja_j(\nu)}{z^j},\qquad
K_\nu(z)\sim\sqrt{\frac\pi{2z}}e^{-z}\sum_{j=0}^\infty\frac{a_j(\nu)}{z^j},\qquad
a_0=1,\quad a_{j+1}=\frac{4\nu^2-(2j+1)^2}{8(j+1)}a_j.
$$
The expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a <differential equation>. These formulas are documented in https://dlmf.nist.gov/10.40[NIST DLMF, equations 10.40.1–2].


 Large-argument asymptotic expansion of a modified Bessel function

ID: large-argument-asymptotic-expansion-of-a-modified-bessel-function