Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. The canonical form of differential equation for these problems is given by

where $z$ is a real or complex variable and $u$ is a large real or complex parameter.

In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_0$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.

For further information and references see §§2.8(i) and 2.8(iv).

In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha$, $f(z,\alpha )$ has a simple zero $z=z_0(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where $z_0(a)=0$. Assume that whether or not $\alpha =a$, $z^{2}g(z,\alpha )$ is analytic at $z=0$. Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. See §2.8(vi) for references.