# §10.72(i) Differential Equations with Turning Points

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

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

# Simple Turning Points

In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, 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 $\tfrac{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.

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

# Multiple or Fractional Turning Points

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

# §10.72(ii) Differential Equations with Poles

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).

# §10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

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.