§2.8 Differential Equations with a Parameter

Many special functions satisfy an equation of the form

2.8.1		$$d^{2}w/{dz}^2=\left(u^{2}f(z)+g(z)\right)w,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $f(x)$: function, $g(x)$: function, $w$: solution and $u$: large real or complex parameter Referenced by: §2.8(i), §2.8(v) Permalink: http://dlmf.nist.gov/2.8.E1 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.2

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 $𝐃$ in $ℝ$ or $ℂ$. 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 $𝐃$, 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 $𝐃$ 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 transformation (§1.13(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, $W$: change of variable and $\dot{z}$: derivative of $z$ with respect to $\xi$ Permalink: http://dlmf.nist.gov/2.8.E2 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.2

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

2.8.3		$$\frac{d^{2}W}{{d\xi }^2}=\left(u^2{\dot{z}}^{2}f(z)+\psi (\xi )\right)W,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $f(x)$: function, $u$: large real or complex parameter, $W$: change of variable, $\dot{z}$: derivative of $z$ with respect to $\xi$ and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E3 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.2

where

2.8.4		$$\psi (\xi )={\dot{z}}^{2}g(z)+{\dot{z}}^{1/2}\frac{d^2}{{d\xi }^2}({\dot{z}}^{-1/2}).$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $g(x)$: function, $\dot{z}$: derivative of $z$ with respect to $\xi$ and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E4 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.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		${\dot{z}}^{2}f(z)$	$=1,$
		$\xi$	$={\displaystyle \int f^{1/2}(z)dz},$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $f(x)$: function and $\dot{z}$: derivative of $z$ with respect to $\xi$ 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 Ch.2

for Case I,

2.8.6		${\dot{z}}^{2}f(z)$	$=\xi ,$
		$\frac{2}{3}{\xi }^{3/2}$	$={\displaystyle {\int }_{z_0}^z}{f}^{1/2}(t)dt,$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $f(x)$: function and $\dot{z}$: derivative of $z$ with respect to $\xi$ 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 Ch.2

for Case II,

2.8.7		${\dot{z}}^{2}f(z)$	$=1/\xi ,$
		$2{\xi }^{1/2}$	$={\displaystyle {\int }_{z_0}^z}{f}^{1/2}(t)dt,$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $f(x)$: function and $\dot{z}$: derivative of $z$ with respect to $\xi$ 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 Ch.2

for Case III.

The transformed equation has the form

2.8.8		$$d^{2}W/{d\xi }^2=\left(u^2{\xi }^m+\psi (\xi )\right)W,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $u$: large real or complex parameter, $W$: change of variable and $\psi (\xi )$: function Referenced by: §2.8(i), §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E8 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.2

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		$$\frac{d^{2}W}{{d\xi }^2}=\left(\frac{u^2}{\xi }+\frac{\rho }{{\xi }^2}\right)W,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $u$: large real or complex parameter, $W$: change of variable and $\rho$: limit Permalink: http://dlmf.nist.gov/2.8.E9 Encodings: TeX, pMML, png See also: Annotations for §2.8(i), §2.8 and Ch.2

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).

The transformed differential equation is

2.8.10		$$d^{2}W/{d\xi }^2=(u^2+\psi (\xi ))W,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $W_{n,j}(u,\xi )$: solution, $u$: large real or complex parameter and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E10 Encodings: TeX, pMML, png See also: Annotations for §2.8(ii), §2.8 and Ch.2

in which $\xi$ ranges over a bounded or unbounded interval or domain $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. 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 $C^{\mathrm{\infty }}$ or analytic on $𝚫$, and as $u\to \mathrm{\infty }$

2.8.11		$W_{n,1}(u,\xi )$	$={\mathrm{e}}^{u\xi }\left({\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^s}}+O\left({\displaystyle \frac{1}{u^n}}\right)\right),$
$\xi \in {𝚫}_1({\alpha }_1)$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter Referenced by: §2.8(ii), §2.8(ii) Permalink: http://dlmf.nist.gov/2.8.E11 Encodings: TeX, pMML, png See also: Annotations for §2.8(ii), §2.8 and Ch.2
2.8.12		$W_{n,2}(u,\xi )$	$={\mathrm{e}}^{-u\xi }\left({\displaystyle \sum _{s=0}^{n-1}}{(-1)}^s{\displaystyle \frac{A_s(\xi )}{u^s}}+O\left({\displaystyle \frac{1}{u^n}}\right)\right),$
$\xi \in {𝚫}_2({\alpha }_2)$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter Referenced by: §2.8(ii), §2.8(ii) Permalink: http://dlmf.nist.gov/2.8.E12 Encodings: TeX, pMML, png See also: Annotations for §2.8(ii), §2.8 and Ch.2

with $A_0(\xi )=1$ and

2.8.13		$$A_{s+1}(\xi )=-\frac{1}{2}{A}_s^{\prime }(\xi )+\frac{1}{2}\int \psi (\xi )A_s(\xi )d\xi ,$$
$s=0,1,2,\mathrm{\dots }$,
ⓘ Symbols: $dx$: 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 Ch.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 ${𝚫}_j({\alpha }_j)$ comprise those points $\xi$ that can be joined to ${\alpha }_j$ in $𝚫$ by a path ${𝒬}_j$ along which $\mathrm{\Re }v$ is nondecreasing $(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from ${\alpha }_j$ to $\xi$. In addition, ${𝒱}_{{𝒬}_j}\left(A_1\right)$ and ${𝒱}_{{𝒬}_j}\left(A_n\right)$ must be bounded on ${𝚫}_j({\alpha }_j)$.

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 $n$.

The transformed differential equation is

2.8.14		$$d^{2}W/{d\xi }^2=(u^2\xi +\psi (\xi ))W,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $W_{n,j}(u,\xi )$: solution, $u$: large real or complex parameter and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E14 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

and for simplicity $\xi$ is assumed to range over a finite or infinite interval $({\alpha }_1,{\alpha }_2)$ with , ${\alpha }_2>0$. Again, $u>0$ and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and as $u\to \mathrm{\infty }$

2.8.15		$W_{n,1}(u,\xi )$	$=\mathrm{Ai}\left(u^{2/3}\xi \right)\left({\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}+O\left({\displaystyle \frac{1}{u^{2n-1}}}\right)\right)+{\mathrm{Ai}}^{\prime }\left(u^{2/3}\xi \right)\left({\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+(4/3)}}}+O\left({\displaystyle \frac{1}{u^{2n-1}}}\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.1(iii), §2.8(iii), §2.8(iii) Permalink: http://dlmf.nist.gov/2.8.E15 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2
2.8.16		$W_{n,2}(u,\xi )$	$=\mathrm{Bi}\left(u^{2/3}\xi \right)\left({\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}+O\left({\displaystyle \frac{1}{u^{2n-1}}}\right)\right)+{\mathrm{Bi}}^{\prime }\left(u^{2/3}\xi \right)\left({\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+(4/3)}}}+O\left({\displaystyle \frac{1}{u^{2n-1}}}\right)\right).$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.8(iii), §2.8(iii) Permalink: http://dlmf.nist.gov/2.8.E16 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

Here $A_0(\xi )=1$,

2.8.17
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E17 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

and

2.8.18		$$A_{s+1}(\xi )=-\frac{1}{2}{B}_s^{\prime }(\xi )+\frac{1}{2}\int \psi (\xi )B_s(\xi )d\xi ,$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E18 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

when $s=0,1,2,\mathrm{\dots }$. 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 ${𝒱}_{{\alpha }_1,{\alpha }_2}\left({|\xi |}^{1/2}{B}_0\right)$ and ${𝒱}_{{\alpha }_1,{\alpha }_2}\left({|\xi |}^{1/2}{B}_{n-1}\right)$ are finite.

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

2.8.19		$$\mathrm{Ai}\left(x\right)=\mathrm{Bi}\left(x\right)$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{Bi}\left(z\right)$: Airy function Permalink: http://dlmf.nist.gov/2.8.E19 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.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		$$\mathrm{envAi}\left(x\right)=\mathrm{envBi}\left(x\right)={\left({\mathrm{Ai}}^2\left(x\right)+{\mathrm{Bi}}^2\left(x\right)\right)}^{1/2},$$
,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ and $c$: root Permalink: http://dlmf.nist.gov/2.8.E20 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

2.8.21		$\mathrm{envAi}\left(x\right)$	$=\sqrt{2}\mathrm{Ai}\left(x\right),$
		$\mathrm{envBi}\left(x\right)$	$=\sqrt{2}\mathrm{Bi}\left(x\right)$,
	.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ and $c$: root Permalink: http://dlmf.nist.gov/2.8.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

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

2.8.22		$W_{n,1}(u,\xi )$	$=\mathrm{Ai}\left(u^{2/3}\xi \right){\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}+{\mathrm{Ai}}^{\prime }\left(u^{2/3}\xi \right){\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+(4/3)}}}+\mathrm{envAi}\left(u^{2/3}\xi \right)O\left({\displaystyle \frac{1}{u^{2n-1}}}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Permalink: http://dlmf.nist.gov/2.8.E22 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2
2.8.23		$W_{n,2}(u,\xi )$	$=\mathrm{Bi}\left(u^{2/3}\xi \right){\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}+{\mathrm{Bi}}^{\prime }\left(u^{2/3}\xi \right){\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+(4/3)}}}+\mathrm{envBi}\left(u^{2/3}\xi \right)O\left({\displaystyle \frac{1}{u^{2n-1}}}\right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Permalink: http://dlmf.nist.gov/2.8.E23 Encodings: TeX, pMML, png See also: Annotations for §2.8(iii), §2.8 and Ch.2

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.

The transformed equation (2.8.8) is renormalized as

2.8.24		$$\frac{d^{2}W}{{d\xi }^2}=\left(\frac{u^2}{4\xi }+\frac{{\nu }^2-1}{4{\xi }^2}+\frac{\psi (\xi )}{\xi }\right)W.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $u$: large real or complex parameter and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E24 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

We again assume $\xi \in ({\alpha }_1,{\alpha }_2)$ with , . Also, $\psi (\xi )$ is $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ on $(0,{\alpha }_2)$, and as $u\to \mathrm{\infty }$

2.8.25		$W_{n,1}(u,\xi )$	$=\begin{array}{l}{\xi }^{1/2}{I}_{\nu }\left(u{\xi }^{1/2}\right){\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}+\xi I_{\nu +1}\left(u{\xi }^{1/2}\right){\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+1}}}\\ +{\xi }^{1/2}{I}_{\nu }\left(u{\xi }^{1/2}\right)O\left({\displaystyle \frac{1}{u^{2n-1}}}\right),\end{array}$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E25 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2
2.8.26		$W_{n,2}(u,\xi )$	$=\begin{array}{l}{\xi }^{1/2}{K}_{\nu }\left(u{\xi }^{1/2}\right){\displaystyle \sum _{s=0}^{n-1}}{\displaystyle \frac{A_s(\xi )}{u^{2s}}}-\xi K_{\nu +1}\left(u{\xi }^{1/2}\right){\displaystyle \sum _{s=0}^{n-2}}{\displaystyle \frac{B_s(\xi )}{u^{2s+1}}}\\ +{\xi }^{1/2}{K}_{\nu }\left(u{\xi }^{1/2}\right)O\left({\displaystyle \frac{1}{u^{2n-1}}}\right).\end{array}$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E26 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

Here $A_0(\xi )=1$,

2.8.27		$$B_s(\xi )=-A_s^{\prime }(\xi )+\frac{1}{{\xi }^{1/2}}{\int }_0^{\xi }\left(\psi (v)A_s(v)-\left(\nu +\frac{1}{2}\right)A_s^{\prime }(v)\right)\frac{dv}{v^{1/2}},$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $\nu$: real nonnegative constant, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E27 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

2.8.28		$$A_{s+1}(\xi )=\nu B_s(\xi )-\xi B_s^{\prime }(\xi )+\int \psi (\xi )B_s(\xi )d\xi ,$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $\nu$: real nonnegative constant, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $\psi (\xi )$: function Referenced by: §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E28 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

$s=0,1,2,\mathrm{\dots }$. 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 ${𝒱}_{0,{\alpha }_2}\left({\xi }^{1/2}{B}_0\right)$ and ${𝒱}_{0,{\alpha }_2}\left({\xi }^{1/2}{B}_{n-1}\right)$ are finite.

If $\xi \in ({\alpha }_1,0)$, then there are solutions $W_{n,j}(u,\xi )$, $j=3,4$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,0)$, and as $u\to \mathrm{\infty }$

2.8.29		$$W_{n,3}(u,\xi )=\begin{array}{l}{|\xi |}^{1/2}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)\\ -|\xi |J_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right),\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.8(iv), §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E29 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

2.8.30		$$W_{n,4}(u,\xi )=\begin{array}{l}{|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)\\ -|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right).\end{array}$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Referenced by: §2.8(iv), §2.8(iv) Permalink: http://dlmf.nist.gov/2.8.E30 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

Here $A_0(\xi )=1$,

2.8.31		$$B_s(\xi )=-A_s^{\prime }(\xi )+\frac{1}{{|\xi |}^{1/2}}{\int }_{\xi }^0\left(\psi (v)A_s(v)-\left(\nu +\frac{1}{2}\right)A_s^{\prime }(v)\right)\frac{dv}{{|v|}^{1/2}},$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $\nu$: real nonnegative constant, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $\psi (\xi )$: function Permalink: http://dlmf.nist.gov/2.8.E31 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

$s=0,1,2,\mathrm{\dots }$, 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 ${𝒱}_{{\alpha }_1,0}\left({|\xi |}^{1/2}{B}_0\right)$ and ${𝒱}_{{\alpha }_1,0}\left({|\xi |}^{1/2}{B}_{n-1}\right)$ 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		$$J_{\nu }(x)+Y_{\nu }(x)=0.$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind and $\nu$: real nonnegative constant Permalink: http://dlmf.nist.gov/2.8.E32 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

Define

2.8.33		$\mathrm{env}{J}_{\nu }(x)$	$=\sqrt{2}{J}_{\nu }(x),$
		$\mathrm{env}{Y}_{\nu }(x)$	$=\sqrt{2}\left|Y_{\nu }(x)\right|$,
	,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\nu$: real nonnegative constant and $X_{\nu }$: smallest positive root Permalink: http://dlmf.nist.gov/2.8.E33 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

2.8.34		$$\mathrm{env}{J}_{\nu }(x)=\mathrm{env}{Y}_{\nu }(x)={\left(J_{\nu }^2(x)+Y_{\nu }^2(x)\right)}^{1/2},$$
.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\nu$: real nonnegative constant and $X_{\nu }$: smallest positive root Permalink: http://dlmf.nist.gov/2.8.E34 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

Then as $u\to \mathrm{\infty }$

2.8.35		$$W_{n,3}(u,\xi )=\begin{array}{l}{|\xi |}^{1/2}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |J_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}\\ +{|\xi |}^{1/2}\mathrm{env}{J}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Permalink: http://dlmf.nist.gov/2.8.E35 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2

2.8.36		$$W_{n,4}(u,\xi )=\begin{array}{l}{|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}\\ +{|\xi |}^{1/2}\mathrm{env}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),\end{array}$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter Permalink: http://dlmf.nist.gov/2.8.E36 Encodings: TeX, pMML, png See also: Annotations for §2.8(iv), §2.8 and Ch.2