§ 2.8. Differential Equations with a Parameter

Permalink:: http://dlmf.nist.gov/2.8

§2.8(i)Classification of Cases
§2.8(ii)Case I: No Transition Points
§2.8(iii)Case II: Simple Turning Point
§2.8(iv)Case III: Simple Pole
§2.8(v)Multiple and Fractional Turning Points
§2.8(vi)Coalescing Transition Points

§ 2.8(i). Classification of Cases

Notes:: See Olver (1997b, pp. 362–363).
Permalink:: http://dlmf.nist.gov/2.8.SS1

Many special functions satisfy an equation of the form

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

Defines:: $f(x)$ : 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
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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 $\Real$ or $\Complex$ . 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 transformation (§Ch.1) to (2.8.1). This introduces new variables $W$ and $\xi$ , related by

2.8.2 $W=\dot{z}^{{-1/2}}w,$

Defines:: $W$ : change of variable
Symbols:: $w$ : solution
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dots denoting differentiations with respect to $\xi$ . Then

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

Defines:: $W$ : change of variable and $\psi(\xi)$ : function
Symbols:: $f(x)$ : function and $u$ : large real or complex parameter
Permalink:: http://dlmf.nist.gov/2.8.E3
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where

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

Defines:: $\psi(\xi)$ : function
Symbols:: $g(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E4
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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

$\dot{z}^{2}f(z)=1,$

$\xi=\int f^{{1/2}}(z)dz,$

Symbols:: $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E5
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for Case I,

2.8.6

$\dot{z}^{2}f(z)=\xi,$

$\tfrac{2}{3}\xi^{{3/2}}=\int _{{z_{0}}}^{{z}}f^{{1/2}}(t)dt,$

Symbols:: $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E6
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for Case II,

2.8.7

$\dot{z}^{2}f(z)=1/\xi,$

$2\xi^{{1/2}}=\int _{{z_{0}}}^{{z}}f^{{1/2}}(t)dt,$

Symbols:: $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E7
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for Case III.

The transformed equation has the form

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

Symbols:: $u$ : 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
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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 $\frac{{d}^{2}W}{{d\xi}^{2}}=\left(\frac{u^{2}}{\xi}+\frac{\rho}{\xi^{2}}\right)W,$

Defines:: $\rho$ : limit
Symbols:: $u$ : large real or complex parameter and $W$ : change of variable
Permalink:: http://dlmf.nist.gov/2.8.E9
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where $\rho=\lim(\xi^{2}\psi(\xi))$ as $\xi\to 0$ . Solutions are Bessel functions, or modified Bessel functions, of order $\pm(1+4\rho)^{{1/2}}$ (§§Ch.10, Ch.10).

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

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

Notes:: See Olver (1997b, pp. 364–368, 371–373).
Permalink:: http://dlmf.nist.gov/2.8.SS2

The transformed differential equation is

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

Defines:: $W_{{n,j}}(u,\xi)$ : solution
Symbols:: $u$ : large real or complex parameter and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E10
Encodings:: TeX, pMathML, png

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 $W_{{n,1}}(u,\xi)=e^{{u\xi}}\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{s}}+O\!\left(\frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{1}(\alpha _{1})$ ,

Defines:: $n$ : positive integer, $W_{{n,j}}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients and $\mathbf{\Delta}_{j}(\alpha _{j})$ : domain
Symbols:: $O\!\left(x\right)$ : order symbol and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E11
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2.8.12 $W_{{n,2}}(u,\xi)=e^{{-u\xi}}\left(\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{A_{s}(\xi)}{u^{s}}+O\!\left(\frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{2}(\alpha _{2})$ ,

Defines:: $n$ : positive integer, $W_{{n,j}}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients and $\mathbf{\Delta}_{j}(\alpha _{j})$ : domain
Symbols:: $O\!\left(x\right)$ : order symbol and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E12
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with $A_{0}(\xi)=1$ and

2.8.13 $A_{{s+1}}(\xi)=-\tfrac{1}{2}A_{s}^{{\prime}}(\xi)+\tfrac{1}{2}\int\psi(\xi)A_{s}(\xi)d\xi,$ $s=0,1,2,\dots$ ,

Defines:: $A_{s}(\xi)$ : coefficients
Symbols:: $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E13
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(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 $\realpart{v}$ 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

Notes:: See Olver (1997b, pp. 392–397 and 408–413).
Referenced by:: §9.15
Permalink:: http://dlmf.nist.gov/2.8.SS3

The transformed differential equation is

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

Defines:: $W_{{n,j}}(u,\xi)$ : solution
Symbols:: $u$ : large real or complex parameter and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E14
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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 $W_{{n,1}}(u,\xi)=\mathrm{Ai}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+O\!\left(\frac{1}{u^{{2n-1}}}\right)\right)+{{\mathrm{Ai}}^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+O\!\left(\frac{1}{u^{{2n-1}}}\right)\right),$

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : solution, $n$ : positive integer and $u$ : large real or complex parameter
Referenced by:: §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E15
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2.8.16 $W_{{n,2}}(u,\xi)=\mathrm{Bi}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+O\!\left(\frac{1}{u^{{2n-1}}}\right)\right)+{{\mathrm{Bi}}^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+O\!\left(\frac{1}{u^{{2n-1}}}\right)\right).$

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : solution, $n$ : positive integer and $u$ : large real or complex parameter
Referenced by:: §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E16
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Here $A_{0}(\xi)=1$ ,

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

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E17
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and

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

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E18
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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:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $\mathrm{Bi}\!\left(z\right)$ : Airy function
Permalink:: http://dlmf.nist.gov/2.8.E19
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of smallest absolute value, and define the envelopes of $\mathrm{Ai}\!\left(x\right)$ and $\mathrm{Bi}\!\left(x\right)$ by

2.8.20 $\mathrm{env}\mathrm{Ai}\!\left(x\right)=\mathrm{env}\mathrm{Bi}\!\left(x\right)=\left({\mathrm{Ai}^{{2}}}\!\left(x\right)+{\mathrm{Bi}^{{2}}}\!\left(x\right)\right)^{{1/2}},$ $-\infty<x\leq c$ ,

Defines:: $c$ : root
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $\mathrm{Bi}\!\left(z\right)$ : Airy function
Permalink:: http://dlmf.nist.gov/2.8.E20
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2.8.21

$\mathrm{env}\mathrm{Ai}\!\left(x\right)=\sqrt{2}\mathrm{Ai}\!\left(x\right),$

$\mathrm{env}\mathrm{Bi}\!\left(x\right)=\sqrt{2}\mathrm{Bi}\!\left(x\right)$ , $c\leq x<\infty$ .

Defines:: $c$ : root
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $\mathrm{Bi}\!\left(z\right)$ : Airy function
Permalink:: http://dlmf.nist.gov/2.8.E21
Encodings:: TeX, TeX, pMathML, pMathML, png, png

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

2.8.22 $W_{{n,1}}(u,\xi)=\mathrm{Ai}\!\left(u^{{2/3}}\xi\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+{{\mathrm{Ai}}^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+\mathrm{env}\mathrm{Ai}\!\left(u^{{2/3}}\xi\right)O\!\left(\frac{1}{u^{{2n-1}}}\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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, pMathML, png

2.8.23 $W_{{n,2}}(u,\xi)=\mathrm{Bi}\!\left(u^{{2/3}}\xi\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+{{\mathrm{Bi}}^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+\mathrm{env}\mathrm{Bi}\!\left(u^{{2/3}}\xi\right)O\!\left(\frac{1}{u^{{2n-1}}}\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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, pMathML, png

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 (1964), 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 §Ch.10 and §§Ch.12, Ch.12; also §§Ch.12, Ch.13, Ch.13, Ch.13, Ch.15, Ch.18, Ch.30, Ch.30, Ch.32, Ch.32, Ch.33, Ch.33, Ch.33, Ch.36, Ch.36.

§ 2.8(iv). Case III: Simple Pole

Notes:: See Olver (1997b, pp. 435–448).
Permalink:: http://dlmf.nist.gov/2.8.SS4

The transformed equation (2.8.8) is renormalized as

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

Defines:: $W_{{n,j}}(u,\xi)$ : solution and $\nu$ : real nonnegative constant
Symbols:: $u$ : large real or complex parameter and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E24
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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 $W_{{n,1}}(u,\xi)=\xi^{{1/2}}I_{{\nu}}\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+\xi I_{{\nu+1}}\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+\xi^{{1/2}}I_{{\nu}}\!\left(u\xi^{{1/2}}\right)O\!\left(\frac{1}{u^{{2n-1}}}\right),$

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : solution, $\nu$ : real nonnegative constant, $n$ : positive integer and $u$ : large real or complex parameter
Referenced by:: §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E25
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2.8.26 $W_{{n,2}}(u,\xi)=\xi^{{1/2}}K_{{\nu}}\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}-\xi K_{{\nu+1}}\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+\xi^{{1/2}}K_{{\nu}}\!\left(u\xi^{{1/2}}\right)O\!\left(\frac{1}{u^{{2n-1}}}\right).$

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : solution, $\nu$ : real nonnegative constant, $n$ : positive integer and $u$ : large real or complex parameter
Referenced by:: §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E26
Encodings:: TeX, pMathML, png

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

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

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\nu$ : real nonnegative constant and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E27
Encodings:: TeX, pMathML, png

2.8.28 $A_{{s+1}}(\xi)=\nu B_{s}(\xi)-\xi B_{s}^{{\prime}}(\xi)+\int\psi(\xi)B_{s}(\xi)d\xi,$

Defines:: $A_{s}(\xi)$ : coefficients and $B_{s}(\xi)$ : coefficients
Symbols:: $\nu$ : real nonnegative constant and $\psi(\xi)$ : function
Referenced by:: §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E28
Encodings:: TeX, pMathML, png

$s=0,1,2,\dots$ . For $I_{{\nu}}$ and $K_{{\nu}}$ see §Ch.10. 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 $W_{{n,3}}(u,\xi)=|\xi|^{{1/2}}J_{{\nu}}\!\left(u|\xi|^{{1/2}}\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+O\!\left(\frac{1}{u^{{2n-1}}}\right)\right)-|\xi|J_{{\nu+1}}\!\left(u|\xi|^{{1/2}}\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+O\!\left(\frac{1}{u^{{2n-2}}}\right)\right),$

Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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, pMathML, png

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

Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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, pMathML, png

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

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

Symbols:: $\nu$ : 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, pMathML, png

$s=0,1,2,\dots$ , and (2.8.28) again applies. For $J_{{\nu}}$ and $Y_{{\nu}}$ see §Ch.10. 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 $J_{{\nu}}(x)+Y_{{\nu}}(x)=0.$

Symbols:: $\nu$ : real nonnegative constant
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Define

2.8.33

$\mathrm{env}J_{{\nu}}(x)=\sqrt{2}J_{{\nu}}(x),$

$\mathrm{env}Y_{{\nu}}(x)=\sqrt{2}\left|Y_{{\nu}}(x)\right|$ , $0<x\leq X_{\nu}$ ,

Defines:: $X_{\nu}$ : smallest positive root
Symbols:: $\nu$ : real nonnegative constant
Permalink:: http://dlmf.nist.gov/2.8.E33
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2.8.34 $\mathrm{env}J_{{\nu}}(x)=\mathrm{env}Y_{{\nu}}(x)=\left({J_{{\nu}}^{{2}}}(x)+{Y_{{\nu}}^{{2}}}(x)\right)^{{1/2}},$ $X_{\nu}\leq x<\infty$ .

Defines:: $X_{\nu}$ : smallest positive root
Symbols:: $\nu$ : real nonnegative constant
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Then as $u\to\infty$

2.8.35 $W_{{n,3}}(u,\xi)=|\xi|^{{1/2}}J_{{\nu}}\!\left(u|\xi|^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}-|\xi|J_{{\nu+1}}\!\left(u|\xi|^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+|\xi|^{{1/2}}\mathrm{env}J_{{\nu}}\!\left(u|\xi|^{{1/2}}\right)O\!\left(\frac{1}{u^{{2n-1}}}\right),$

Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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
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2.8.36 $W_{{n,4}}(u,\xi)=|\xi|^{{1/2}}Y_{{\nu}}\!\left(u|\xi|^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}-|\xi|Y_{{\nu+1}}\!\left(u|\xi|^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+|\xi|^{{1/2}}\mathrm{env}Y_{{\nu}}\!\left(u|\xi|^{{1/2}}\right)O\!\left(\frac{1}{u^{{2n-1}}}\right),$

Symbols:: $O\!\left(x\right)$ : order symbol, $W_{{n,j}}(u,\xi)$ : 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
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uniformly with respect to $\xi\in(\alpha _{1},0)$ .

For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$ , zeros, and examples see Olver (1997b, Chapter 12) and Boyd (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 §§Ch.13, Ch.13, Ch.13, Ch.14, Ch.14, Ch.14, Ch.15, Ch.18, Ch.18, Ch.18, Ch.33.

§ 2.8(v). Multiple and Fractional Turning Points

Referenced by:: §9.13(i)
Permalink:: http://dlmf.nist.gov/2.8.SS5

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 (1977a, b, 1978).

§ 2.8(vi). Coalescing Transition Points

Permalink:: http://dlmf.nist.gov/2.8.SS6

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 (1975, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §Ch.14).

For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990); 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 (1994); in this case the uniform approximants are Whittaker functions (§Ch.13) with a fixed value of the second parameter.

For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§Ch.13, Ch.13, Ch.14, Ch.15, Ch.18.

For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§Ch.13, Ch.14, Ch.14, Ch.14, Ch.30, Ch.30.

For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §Ch.18 and §Ch.28.

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

§ 2.8. Differential Equations with a Parameter

Contents

§ 2.8(i). Classification of Cases

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

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

§ 2.8(iv). Case III: Simple Pole

§ 2.8(v). Multiple and Fractional Turning Points

§ 2.8(vi). Coalescing Transition Points