§2.8 Differential Equations with a Parameter

Keywords:: asymptotic solutions of differential equations
Referenced by:: ¶ ‣ §28.8(iv)
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).
Keywords:: Liouville transformation for differential equations, asymptotic solutions of differential equations, transition points, turning points
Referenced by:: ¶ ‣ §10.72(i), §10.72(ii), §33.2(i)
Permalink:: http://dlmf.nist.gov/2.8.i

Many special functions satisfy an equation of the form

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

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

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 (§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 and $W$ : change of variable
Permalink:: http://dlmf.nist.gov/2.8.E2
Encodings:: TeX, pMML, png

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 and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E3
Encodings:: TeX, pMML, png

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 and $\psi(\xi)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E4
Encodings:: TeX, pMML, png

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:: $dx$ : differential of $x$ , $\int$ : integral and $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

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:: $dx$ : differential of $x$ , $\int$ : integral and $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

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:: $dx$ : differential of $x$ , $\int$ : integral and $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.8.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

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:: $\frac{df}{dx}$ : 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

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

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

Notes:: See Olver (1997b, pp. 364–368, 371–373).
Keywords:: asymptotic solutions of differential equations
Referenced by:: §12.10(i), ¶ ‣ §28.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.ii

The transformed differential equation is

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

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

in which $\xi$ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$ , and $\psi(\xi)$ is $\mathop{C^{{\infty}}\/}\nolimits$ 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 $\mathop{C^{{\infty}}\/}\nolimits$ or analytic on $\mathbf{\Delta}$ , and as $u\to\infty$

2.8.11 $W_{{n,1}}(u,\xi)=e^{{u\xi}}\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{s}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{1}(\alpha _{1})$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\in$ : element of, $e$ : base of exponential function, $n$ : positive integer, $W_{{n,j}}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha _{j})$ : 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

2.8.12 $W_{{n,2}}(u,\xi)=e^{{-u\xi}}\left(\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{A_{s}(\xi)}{u^{s}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{n}}\right)\right),$ $\xi\in\mathbf{\Delta}_{2}(\alpha _{2})$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\in$ : element of, $e$ : base of exponential function, $n$ : positive integer, $W_{{n,j}}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha _{j})$ : 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

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

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

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

(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, $\mathop{\mathcal{V}_{{\mathscr{Q}_{j}}}\/}\nolimits\!\left(A_{1}\right)$ and $\mathop{\mathcal{V}_{{\mathscr{Q}_{j}}}\/}\nolimits\!\left(A_{n}\right)$ 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).
Keywords:: Airy functions, asymptotic solutions of differential equations
Referenced by:: ¶ ‣ §10.72(i), §12.10(i), §13.21(iii), ¶ ‣ §18.15(iv), §33.12(ii), §9.15
Permalink:: http://dlmf.nist.gov/2.8.iii

The transformed differential equation is

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

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

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 $\mathop{C^{{\infty}}\/}\nolimits$ on $(\alpha _{1},\alpha _{2})$ . Corresponding to each positive integer $n$ there are solutions $W_{{n,j}}(u,\xi)$ , $j=1,2$ , that are $\mathop{C^{{\infty}}\/}\nolimits$ on $(\alpha _{1},\alpha _{2})$ , and as $u\to\infty$

2.8.15 $W_{{n,1}}(u,\xi)=\mathop{\mathrm{Ai}\/}\nolimits\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right)\right)+{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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.E15
Encodings:: TeX, pMML, png

2.8.16 $W_{{n,2}}(u,\xi)=\mathop{\mathrm{Bi}\/}\nolimits\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right)\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(u^{{2/3}}\xi\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+(4/3)}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right)\right).$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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

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

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

and

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

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

when $s=0,1,2,\dots$ . For $\mathop{\mathrm{Ai}\/}\nolimits$ and $\mathop{\mathrm{Bi}\/}\nolimits$ 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 $\mathop{\mathcal{V}_{{\alpha _{1},\alpha _{2}}}\/}\nolimits\!\left(|\xi|^{{1/2}}B_{0}\right)$ and $\mathop{\mathcal{V}_{{\alpha _{1},\alpha _{2}}}\/}\nolimits\!\left(|\xi|^{{1/2}}B_{{n-1}}\right)$ 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 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function
Permalink:: http://dlmf.nist.gov/2.8.E19
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function and $c$ : root
Permalink:: http://dlmf.nist.gov/2.8.E20
Encodings:: TeX, pMML, png

2.8.21

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

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

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

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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
Permalink:: http://dlmf.nist.gov/2.8.E22
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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
Permalink:: http://dlmf.nist.gov/2.8.E23
Encodings:: TeX, pMML, 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 (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

Notes:: See Olver (1997b, pp. 435–448).
Keywords:: Bessel functions, asymptotic solutions of differential equations
Referenced by:: §10.72(ii), §13.21(i), §13.21(ii), §13.8(iii), §14.15(iii), ¶ ‣ §18.15(iv), §33.12(ii)
Permalink:: http://dlmf.nist.gov/2.8.iv

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

We again assume $\xi\in(\alpha _{1},\alpha _{2})$ with $-\infty\leq\alpha _{1}<0$ , $0<\alpha _{2}\leq\infty$ . Also, $\psi(\xi)$ is $\mathop{C^{{\infty}}\/}\nolimits$ 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 $\mathop{C^{{\infty}}\/}\nolimits$ on $(0,\alpha _{2})$ , and as $u\to\infty$

2.8.25 $W_{{n,1}}(u,\xi)=\xi^{{1/2}}\mathop{I_{{\nu}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+\xi\mathop{I_{{\nu+1}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+\xi^{{1/2}}\mathop{I_{{\nu}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $W_{{n,j}}(u,\xi)$ : 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

2.8.26 $W_{{n,2}}(u,\xi)=\xi^{{1/2}}\mathop{K_{{\nu}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}-\xi\mathop{K_{{\nu+1}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+\xi^{{1/2}}\mathop{K_{{\nu}}\/}\nolimits\!\left(u\xi^{{1/2}}\right)\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $W_{{n,j}}(u,\xi)$ : 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

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

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

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

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

$s=0,1,2,\dots$ . For $\mathop{I_{{\nu}}\/}\nolimits$ and $\mathop{K_{{\nu}}\/}\nolimits$ 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 $\mathop{\mathcal{V}_{{0,\alpha _{2}}}\/}\nolimits\!\left(\xi^{{1/2}}B_{0}\right)$ and $\mathop{\mathcal{V}_{{0,\alpha _{2}}}\/}\nolimits\!\left(\xi^{{1/2}}B_{{n-1}}\right)$ are finite.

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

2.8.29 $W_{{n,3}}(u,\xi)=|\xi|^{{1/2}}\mathop{J_{{\nu}}\/}\nolimits\!\left(u|\xi|^{{1/2}}\right)\left(\sum _{{s=0}}^{{n-1}}\frac{A_{s}(\xi)}{u^{{2s}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-1}}}\right)\right)-|\xi|\mathop{J_{{\nu+1}}\/}\nolimits\!\left(u|\xi|^{{1/2}}\right)\left(\sum _{{s=0}}^{{n-2}}\frac{B_{s}(\xi)}{u^{{2s+1}}}+\mathop{O\/}\nolimits\!\left(\frac{1}{u^{{2n-2}}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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:: $dx$ : 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

$s=0,1,2,\dots$ , and (2.8.28) again applies. For $\mathop{J_{{\nu}}\/}\nolimits$ and $\mathop{Y_{{\nu}}\/}\nolimits$ 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 $\mathop{\mathcal{V}_{{\alpha _{1},0}}\/}\nolimits\!\left(|\xi|^{{1/2}}B_{0}\right)$ and $\mathop{\mathcal{V}_{{\alpha _{1},0}}\/}\nolimits\!\left(|\xi|^{{1/2}}B_{{n-1}}\right)$ 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 $\mathop{J_{{\nu}}\/}\nolimits(x)+\mathop{Y_{{\nu}}\/}\nolimits(x)=0.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind and $\nu$ : real nonnegative constant
Permalink:: http://dlmf.nist.gov/2.8.E32
Encodings:: TeX, pMML, png

Define

2.8.33

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\nu$ : real nonnegative constant and $X_{\nu}$ : smallest positive root
Permalink:: http://dlmf.nist.gov/2.8.E33
Encodings:: TeX, TeX, pMML, pMML, png, png

2.8.34 $\mathop{\mathrm{env}\/}\nolimits\mathop{J_{{\nu}}\/}\nolimits(x)=\mathop{\mathrm{env}\/}\nolimits\mathop{Y_{{\nu}}\/}\nolimits(x)=\left({\mathop{J_{{\nu}}\/}\nolimits^{{2}}}(x)+{\mathop{Y_{{\nu}}\/}\nolimits^{{2}}}(x)\right)^{{1/2}},$ $X_{\nu}\leq x<\infty$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\nu$ : real nonnegative constant and $X_{\nu}$ : smallest positive root
Permalink:: http://dlmf.nist.gov/2.8.E34
Encodings:: TeX, pMML, png

Then as $u\to\infty$

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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
Permalink:: http://dlmf.nist.gov/2.8.E35
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : 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
Permalink:: http://dlmf.nist.gov/2.8.E36
Encodings:: TeX, pMML, png

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

Keywords:: asymptotic solutions of differential equations, turning points
Referenced by:: ¶ ‣ §10.72(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/2.8.v

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

Keywords:: asymptotic solutions of differential equations
Referenced by:: §10.72(iii), §12.16, §13.27, ¶ ‣ §28.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.vi

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).

§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