# §2.8 Differential Equations with a Parameter

## §2.8(i) Classification of Cases

Many special functions satisfy an equation of the form

 2.8.1 $\ifrac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(u^{2}f(z)+g(z)\right)w,$

in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $\mathbf{D}$ in $\mathbb{R}$ or $\mathbb{C}$. For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. The form of the asymptotic expansion depends on the nature of the transition points in $\mathbf{D}$, that is, points at which $f(z)$ has a zero or singularity. Zeros of $f(z)$ are also called turning points.

There are three main cases. In Case I there are no transition points in $\mathbf{D}$ and $g(z)$ is analytic. In Case II $f(z)$ has a simple zero at $z_{0}$ and $g(z)$ is analytic at $z_{0}$. In Case III $f(z)$ has a simple pole at $z_{0}$ and $(z-z_{0})^{2}g(z)$ is analytic at $z_{0}$.

The same approach is used in all three cases. First we apply the Liouville transformation1.13(iv)) to (2.8.1). This introduces new variables $W$ and $\xi$, related by

 2.8.2 $W=\dot{z}^{-1/2}w,$ ⓘ Symbols: $w$: solution and $W$: change of variable Permalink: http://dlmf.nist.gov/2.8.E2 Encodings: TeX, pMML, png See also: Annotations for 2.8(i), 2.8 and 2

dots denoting differentiations with respect to $\xi$. Then

 2.8.3 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(u^{2}\dot{z}^{2}f(z)+\psi(% \xi)\right)W,$

where

 2.8.4 $\psi(\xi)=\dot{z}^{2}g(z)+\dot{z}^{1/2}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}\xi}% ^{2}}(\dot{z}^{-1/2}).$

The transformation is now specialized in such a way that: (a) $\xi$ and $z$ are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting $\psi(\xi)$ (or part of $\psi(\xi)$) has solutions that are functions of a single variable. The actual choices are as follows:

 2.8.5 $\displaystyle\dot{z}^{2}f(z)$ $\displaystyle=1,$ $\displaystyle\xi$ $\displaystyle=\int f^{1/2}(z)\mathrm{d}z,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $f(x)$: function Permalink: http://dlmf.nist.gov/2.8.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 2.8(i), 2.8 and 2

for Case I,

 2.8.6 $\displaystyle\dot{z}^{2}f(z)$ $\displaystyle=\xi,$ $\displaystyle\tfrac{2}{3}\xi^{3/2}$ $\displaystyle=\int_{z_{0}}^{z}f^{1/2}(t)\mathrm{d}t,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $f(x)$: function Permalink: http://dlmf.nist.gov/2.8.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 2.8(i), 2.8 and 2

for Case II,

 2.8.7 $\displaystyle\dot{z}^{2}f(z)$ $\displaystyle=1/\xi,$ $\displaystyle 2\xi^{1/2}$ $\displaystyle=\int_{z_{0}}^{z}f^{1/2}(t)\mathrm{d}t,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $f(x)$: function Permalink: http://dlmf.nist.gov/2.8.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 2.8(i), 2.8 and 2

for Case III.

The transformed equation has the form

 2.8.8 $\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(u^{2}\xi^{m}+\psi(\xi)% \right)W,$

with $m=0$ (Case I), $m=1$ (Case II), $m=-1$ (Case III). In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with $\psi(\xi)=0$. These are elementary functions in Case I, and Airy functions (§9.2) in Case II. In Case III the approximating equation is

 2.8.9 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(\frac{u^{2}}{\xi}+\frac{% \rho}{\xi^{2}}\right)W,$

where $\rho=\lim(\xi^{2}\psi(\xi))$ as $\xi\to 0$. Solutions are Bessel functions, or modified Bessel functions, of order $\pm(1+4\rho)^{1/2}$ (§§10.2, 10.25).

For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004).

## §2.8(ii) Case I: No Transition Points

The transformed differential equation is

 2.8.10 $\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=(u^{2}+\psi(\xi))W,$

in which $\xi$ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$, and $\psi(\xi)$ is $C^{\infty}$ or analytic on $\mathbf{\Delta}$. The parameter $u$ is assumed to be real and positive. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that depend on arbitrarily chosen reference points $\alpha_{j}$, are $C^{\infty}$ or analytic on $\mathbf{\Delta}$, and as $u\to\infty$

 2.8.11 $\displaystyle W_{n,1}(u,\xi)$ $\displaystyle=e^{u\xi}\left(\sum_{s=0}^{n-1}\frac{A_{s}(\xi)}{u^{s}}+O\left(% \frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{1}(\alpha_{1})$, 2.8.12 $\displaystyle W_{n,2}(u,\xi)$ $\displaystyle=e^{-u\xi}\left(\sum_{s=0}^{n-1}(-1)^{s}\frac{A_{s}(\xi)}{u^{s}}+% O\left(\frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{2}(\alpha_{2})$,

with $A_{0}(\xi)=1$ and

 2.8.13 $A_{s+1}(\xi)=-\tfrac{1}{2}A_{s}^{\prime}(\xi)+\tfrac{1}{2}\int\psi(\xi)A_{s}(% \xi)\mathrm{d}\xi,$ $s=0,1,2,\dots$, ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $A_{s}(\xi)$: coefficients and $\psi(\xi)$: function Permalink: http://dlmf.nist.gov/2.8.E13 Encodings: TeX, pMML, png See also: Annotations for 2.8(ii), 2.8 and 2

(the constants of integration being arbitrary). The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. The regions of validity $\mathbf{\Delta}_{j}(\alpha_{j})$ comprise those points $\xi$ that can be joined to $\alpha_{j}$ in $\mathbf{\Delta}$ by a path $\mathscr{Q}_{j}$ along which $\Re v$ is nondecreasing $(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from $\alpha_{j}$ to $\xi$. In addition, $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{1}\right)$ and $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{n}\right)$ must be bounded on $\mathbf{\Delta}_{j}(\alpha_{j})$.

For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). This reference also supplies sufficient conditions to ensure that the solutions $W_{n,1}(u,\xi)$ and $W_{n,2}(u,\xi)$ having the properties (2.8.11) and (2.8.12) are independent of $n$.

## §2.8(iii) Case II: Simple Turning Point

The transformed differential equation is

 2.8.14 $\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=(u^{2}\xi+\psi(\xi))W,$

and for simplicity $\xi$ is assumed to range over a finite or infinite interval $(\alpha_{1},\alpha_{2})$ with $\alpha_{1}<0$, $\alpha_{2}>0$. Again, $u>0$ and $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and as $u\to\infty$

 2.8.15 $\displaystyle W_{n,1}(u,\xi)$ $\displaystyle=\mathrm{Ai}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}\frac{A_% {s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+\mathrm{Ai}'\left(u^% {2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+(4/3)}}+O\left(% \frac{1}{u^{2n-1}}\right)\right),$ 2.8.16 $\displaystyle W_{n,2}(u,\xi)$ $\displaystyle=\mathrm{Bi}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}\frac{A_% {s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+\mathrm{Bi}'\left(u^% {2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+(4/3)}}+O\left(% \frac{1}{u^{2n-1}}\right)\right).$

Here $A_{0}(\xi)=1$,

 2.8.17 $B_{s}(\xi)=\begin{cases}\dfrac{1}{2\xi^{1/2}}\displaystyle\int_{0}^{\xi}\left(% \psi(v)A_{s}(v)-A_{s}^{\prime\prime}(v)\right)\dfrac{\mathrm{d}v}{v^{1/2}},&% \xi>0,\\ \dfrac{1}{2(-\xi)^{1/2}}\displaystyle\int_{\xi}^{0}\left(\psi(v)A_{s}(v)-A_{s}% ^{\prime\prime}(v)\right)\dfrac{\mathrm{d}v}{(-v)^{1/2}},&\xi<0,\end{cases}$

and

 2.8.18 $A_{s+1}(\xi)=-\tfrac{1}{2}B_{s}^{\prime}(\xi)+\tfrac{1}{2}\int\psi(\xi)B_{s}(% \xi)\mathrm{d}\xi,$

when $s=0,1,2,\dots$. For $\mathrm{Ai}$ and $\mathrm{Bi}$ see §9.2. The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. These results are valid when $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite.

An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. Let $c=-0.36604...$ be the real root of the equation

 2.8.19 $\mathrm{Ai}\left(x\right)=\mathrm{Bi}\left(x\right)$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function and $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function Permalink: http://dlmf.nist.gov/2.8.E19 Encodings: TeX, pMML, png See also: Annotations for 2.8(iii), 2.8 and 2

of smallest absolute value, and define the envelopes of $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ by

 2.8.20 $\mathrm{envAi}\left(x\right)=\mathrm{envBi}\left(x\right)=\left({\mathrm{Ai}^{% 2}}\left(x\right)+{\mathrm{Bi}^{2}}\left(x\right)\right)^{1/2},$ $-\infty,
 2.8.21 $\displaystyle\mathrm{envAi}\left(x\right)$ $\displaystyle=\sqrt{2}\mathrm{Ai}\left(x\right),$ $\displaystyle\mathrm{envBi}\left(x\right)$ $\displaystyle=\sqrt{2}\mathrm{Bi}\left(x\right)$, $c\leq x<\infty$.

These envelopes are continuous functions of $x$, and as $u\to\infty$

 2.8.22 $\displaystyle W_{n,1}(u,\xi)$ $\displaystyle=\mathrm{Ai}\left(u^{2/3}\xi\right)\sum_{s=0}^{n-1}\frac{A_{s}(% \xi)}{u^{2s}}+\mathrm{Ai}'\left(u^{2/3}\xi\right)\sum_{s=0}^{n-2}\frac{B_{s}(% \xi)}{u^{2s+(4/3)}}+\mathrm{envAi}\left(u^{2/3}\xi\right)O\left(\frac{1}{u^{2n% -1}}\right),$ 2.8.23 $\displaystyle W_{n,2}(u,\xi)$ $\displaystyle=\mathrm{Bi}\left(u^{2/3}\xi\right)\sum_{s=0}^{n-1}\frac{A_{s}(% \xi)}{u^{2s}}+\mathrm{Bi}'\left(u^{2/3}\xi\right)\sum_{s=0}^{n-2}\frac{B_{s}(% \xi)}{u^{2s+(4/3)}}+\mathrm{envBi}\left(u^{2/3}\xi\right)O\left(\frac{1}{u^{2n% -1}}\right),$

uniformly with respect to $\xi\in(\alpha_{1},\alpha_{2})$.

For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13.

## §2.8(iv) Case III: Simple Pole

The transformed equation (2.8.8) is renormalized as

 2.8.24 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(\frac{u^{2}}{4\xi}+\frac{% \nu^{2}-1}{4\xi^{2}}+\frac{\psi(\xi)}{\xi}\right)W.$

We again assume $\xi\in(\alpha_{1},\alpha_{2})$ with $-\infty\leq\alpha_{1}<0$, $0<\alpha_{2}\leq\infty$. Also, $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and $u>0$. The constant $\nu$ ($=\sqrt{1+4\rho}$) is real and nonnegative.

There are two cases: $\xi\in(0,\alpha_{2})$ and $\xi\in(\alpha_{1},0)$. In the former, corresponding to any positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(0,\alpha_{2})$, and as $u\to\infty$

 2.8.25 $\displaystyle W_{n,1}(u,\xi)$ $\displaystyle=\xi^{1/2}I_{\nu}\left(u\xi^{1/2}\right)\sum_{s=0}^{n-1}\frac{A_{% s}(\xi)}{u^{2s}}+\xi I_{\nu+1}\left(u\xi^{1/2}\right)\sum_{s=0}^{n-2}\frac{B_{% s}(\xi)}{u^{2s+1}}+\xi^{1/2}I_{\nu}\left(u\xi^{1/2}\right)O\left(\frac{1}{u^{2% n-1}}\right),$ 2.8.26 $\displaystyle W_{n,2}(u,\xi)$ $\displaystyle=\xi^{1/2}K_{\nu}\left(u\xi^{1/2}\right)\sum_{s=0}^{n-1}\frac{A_{% s}(\xi)}{u^{2s}}-\xi K_{\nu+1}\left(u\xi^{1/2}\right)\sum_{s=0}^{n-2}\frac{B_{% s}(\xi)}{u^{2s+1}}+\xi^{1/2}K_{\nu}\left(u\xi^{1/2}\right)O\left(\frac{1}{u^{2% n-1}}\right).$

Here $A_{0}(\xi)=1$,

 2.8.27 $B_{s}(\xi)=-A_{s}^{\prime}(\xi)+\frac{1}{\xi^{1/2}}\int_{0}^{\xi}\left(\psi(v)% A_{s}(v)-\left(\nu+\tfrac{1}{2}\right)A_{s}^{\prime}(v)\right)\frac{\mathrm{d}% v}{v^{1/2}},$
 2.8.28 $A_{s+1}(\xi)=\nu B_{s}(\xi)-\xi B_{s}^{\prime}(\xi)+\int\psi(\xi)B_{s}(\xi)% \mathrm{d}\xi,$

$s=0,1,2,\dots$. For $I_{\nu}$ and $K_{\nu}$ see §10.25(ii). The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. These results are valid when $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{0}\right)$ and $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{n-1}\right)$ are finite.

If $\xi\in(\alpha_{1},0)$, then there are solutions $W_{n,j}(u,\xi)$, $j=3,4$, that are $C^{\infty}$ on $(\alpha_{1},0)$, and as $u\to\infty$

 2.8.29 $W_{n,3}(u,\xi)=|\xi|^{1/2}J_{\nu}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-% 1}\frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi|J_{% \nu+1}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1% }}+O\left(\frac{1}{u^{2n-2}}\right)\right),$
 2.8.30 $W_{n,4}(u,\xi)=|\xi|^{1/2}Y_{\nu}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-% 1}\frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi|Y_{% \nu+1}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1% }}+O\left(\frac{1}{u^{2n-2}}\right)\right).$

Here $A_{0}(\xi)=1$,

 2.8.31 $B_{s}(\xi)=-A_{s}^{\prime}(\xi)+\frac{1}{|\xi|^{1/2}}\int_{\xi}^{0}\left(\psi(% v)A_{s}(v)-\left(\nu+\tfrac{1}{2}\right)A_{s}^{\prime}(v)\right)\frac{\mathrm{% d}v}{|v|^{1/2}},$

$s=0,1,2,\dots$, and (2.8.28) again applies. For $J_{\nu}$ and $Y_{\nu}$ see §10.2(ii). The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. These results are valid when $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite.

Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. Let $x=X_{\nu}$ be the smallest positive root of the equation

 2.8.32 $J_{\nu}(x)+Y_{\nu}(x)=0.$

Define

 2.8.33 $\displaystyle\mathrm{env}\mskip-2.0mu J_{\nu}(x)$ $\displaystyle=\sqrt{2}J_{\nu}(x),$ $\displaystyle\mathrm{env}\mskip-2.0mu Y_{\nu}(x)$ $\displaystyle=\sqrt{2}\left|Y_{\nu}(x)\right|$, $0,
 2.8.34 $\mathrm{env}\mskip-2.0mu J_{\nu}(x)=\mathrm{env}\mskip-2.0mu Y_{\nu}(x)=\left(% {J_{\nu}^{2}}(x)+{Y_{\nu}^{2}}(x)\right)^{1/2},$ $X_{\nu}\leq x<\infty$.

Then as $u\to\infty$

 2.8.35 $W_{n,3}(u,\xi)=|\xi|^{1/2}J_{\nu}\left(u|\xi|^{1/2}\right)\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}-|\xi|J_{\nu+1}\left(u|\xi|^{1/2}\right)\sum_{s=0}^{n% -2}\frac{B_{s}(\xi)}{u^{2s+1}}+|\xi|^{1/2}\mathrm{env}\mskip-2.0mu J_{\nu}% \left(u|\xi|^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$
 2.8.36 $W_{n,4}(u,\xi)=|\xi|^{1/2}Y_{\nu}\left(u|\xi|^{1/2}\right)\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}-|\xi|Y_{\nu+1}\left(u|\xi|^{1/2}\right)\sum_{s=0}^{n% -2}\frac{B_{s}(\xi)}{u^{2s+1}}+|\xi|^{1/2}\mathrm{env}\mskip-2.0mu Y_{\nu}% \left(u|\xi|^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$

uniformly with respect to $\xi\in(\alpha_{1},0)$.

For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).

## §2.8(v) Multiple and Fractional Turning Points

The approach used in preceding subsections for equation (2.8.1) also succeeds when $z_{0}$ is a multiple or fractional turning point. For the former $f(z)$ has a zero of multiplicity $\lambda=2,3,4,\dots$ and $g(z)$ is analytic. For the latter $(z-z_{0})^{-\lambda}f(z)$ and $g(z)$ are both analytic at $z_{0}$, $\lambda$ ($>-2$) being a real constant. In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order $1/(\lambda+2)$. More generally, $g(z)$ can have a simple or double pole at $z_{0}$. (In the case of the double pole the order of the approximating Bessel functions is fixed but no longer $1/(\lambda+2)$.) However, in all cases with $\lambda>-2$ and $\lambda\neq 0$ or $\pm 1$, only uniform asymptotic approximations are available, not uniform asymptotic expansions. For results, including error bounds, see Olver (1977c).

For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).

## §2.8(vi) Coalescing Transition Points

Corresponding to the problems for integrals outlined in §§2.3(v), 2.4(v), and 2.4(vi), there are analogous problems for differential equations.

For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §14.15(v)).

For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order.

For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter.

For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.

For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii).

For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv).

Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007).