- §10.72(i) Differential Equations with Turning Points
- §10.72(ii) Differential Equations with Poles
- §10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

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

10.72.1 | $$\frac{{d}^{2}w}{{dz}^{2}}=\left({u}^{2}f\left(z\right)+g\left(z\right)\right)w,$$ | ||

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

In regions in which (10.72.1) has a simple turning point ${z}_{0}$, that is, $f\left(z\right)$ and $g\left(z\right)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and ${z}_{0}$ is a simple zero of $f\left(z\right)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). These expansions are uniform with respect to $z$, including the turning point ${z}_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.

If $f\left(z\right)$ has a double zero ${z}_{0}$, or more generally ${z}_{0}$ is a zero of order
$m$, $m=2,3,4,\mathrm{\dots}$, then uniform asymptotic approximations (but
*not* expansions) can be constructed in terms of Bessel functions,
or modified Bessel functions, of order $1/\left(m+2\right)$. The number $m$ can also be
replaced by any real constant $\lambda $ $\left(>-2\right)$ in the sense that
${\left(z-{z}_{0}\right)}^{-\lambda}$ $f\left(z\right)$ is analytic and nonvanishing at ${z}_{0}$; moreover,
$g\left(z\right)$ is permitted to have a single or double pole at ${z}_{0}$. The order of
the approximating Bessel functions, or modified Bessel functions, is
$1/\left(\lambda +2\right)$, except in the case when $g\left(z\right)$ has a double pole at ${z}_{0}$.
See §2.8(v) for references.

In regions in which the function $f\left(z\right)$ has a simple pole at $z={z}_{0}$ and ${\left(z-{z}_{0}\right)}^{2}g\left(z\right)$ 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 ${\left(z-{z}_{0}\right)}^{2}g\left(z\right)$ 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.

In (10.72.1) assume $f\left(z\right)=f\left(z,\alpha \right)$ and $g\left(z\right)=g\left(z,\alpha \right)$ depend continuously on a real parameter $\alpha $, $f\left(z,\alpha \right)$ has a simple zero $z={z}_{0}\left(\alpha \right)$ and a double pole $z=0$, except for a critical value $\alpha =a$, where ${z}_{0}\left(a\right)=0$. Assume that whether or not $\alpha =a$, ${z}^{2}g\left(z,\alpha \right)$ 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 $z={z}_{0}\left(a\right)$, the cut neighborhood of $z=0$, and $\alpha =a$. See §2.8(vi) for references.