asymptotic solutions of difference equations

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21: 9.13 Generalized Airy Functions

… ►are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). … ►As

z\to\infty

… ►In Olver (1977a, 1978) a different normalization is used. … ►Properties and graphs of

U_{m}(t)

V_{m}(t)

\overline{V}_{m}(t)

are included in Olver (1977a) together with properties and graphs of real solutions of the equation … ►and the difference equation …

22: Bibliography G

… ►

W. Gautschi (1997b) The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:: ISBN 90-5699-521-9, MathReview, ZentralBlatt
Referenced by:: §3.6(iii), §3.6(vii).
Encodings:: BibTeX

… ►

V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:: In Russian. English translation: Differential Equations 18(3), pp. 317–326.
External Links:: ISSN 0374-0641, MathReview, ZentralBlatt
Referenced by:: §32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).
Encodings:: BibTeX

►

V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

►

V. I. Gromak (1976) The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:: In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(v).
Encodings:: BibTeX

►

V. I. Gromak (1978) One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).

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Notes:: In Russian. English translation: Differential Equations 14(12), pp. 1510–1513 (1979)
External Links:: ISSN 0374-0641, MathReview (M. Tvrdý), ZentralBlatt
Referenced by:: §32.10(i), §32.10(ii), §32.10(iii), §32.7(iv).
Encodings:: BibTeX

…

23: 2.11 Remainder Terms; Stokes Phenomenon

… ► … ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector

\frac{1}{2}\pi<\operatorname{ph}z<\frac{3}{2}\pi

. … ►

§2.11(v) Exponentially-Improved Expansions (continued)

… ►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … ► …

24: 2.7 Differential Equations

… ►

§2.7(ii) Irregular Singularities of Rank 1

… ►See §2.11(v) for other examples. … ►

§2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … ►

§2.7(iv) Numerically Satisfactory Solutions

…

25: 30.8 Expansions in Series of Ferrers Functions

… ►

30.8.1

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-R}^{\infty}(-1)^{k}a^{m}% _{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right),

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $R=\left\lfloor(n-m)/2\right\rfloor$ : limit and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
A&S Ref:: 21.7.1 (in different form)
Referenced by:: §30.16(ii), §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §30.8(i), §30.8 and Ch.30

… ►Then the set of coefficients

a^{m}_{n,k}(\gamma^{2})

k=-R,-R+1,-R+2,\dots

is the solution of the difference equation ►

30.8.4

A_{k}f_{k-1}+\left(B_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right)\right)f_{k}+C_% {k}f_{k+1}=0,

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Symbols:: $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $A_{k}$ , $B_{k}$ and $C_{k}$
Referenced by:: §30.16(ii), §30.8(ii)
Permalink:: http://dlmf.nist.gov/30.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.9

\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-\infty}^{-N-1}(-1)^{k}{a% ^{\prime}}^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right)+\sum_{k=-N% }^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{Q}^{m}_{n+2k}\left(x\right),

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{n,k}(\gamma^{2})$ : coefficients and $N$ : upper limit
A&S Ref:: 21.7.21 (in different form)
Permalink:: http://dlmf.nist.gov/30.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §30.8(ii), §30.8 and Ch.30

… ►The set of coefficients

{a^{\prime}}^{m}_{n,k}(\gamma^{2})

k=-N-1,-N-2,\dots

, is the recessive solution of (30.8.4) as

k\to-\infty

that is normalized by …

B. R. Fabijonas, D. W. Lozier, and F. W. J. Olver (2004) Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §9.17(ii), §9.17(v), §9.8(iv).
Encodings:: BibTeX

… ►

A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:: ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.13(i), §32.7(i), §32.7(iii).
Encodings:: BibTeX

… ►

A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:: ISSN 0024-1318, MathReview (D. A. Lutz)
Referenced by:: §32.10(vi).
Encodings:: BibTeX

… ►

T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:: MathReview (W. J. Trjitzinsky), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

F. N. Fritsch, R. E. Shafer, and W. P. Crowley (1973) Solution of the transcendental equation

we^{w}=x

. Comm. ACM 16 (2), pp. 123–124.

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Notes:: Includes Fortran code.
External Links:: ISSN 0001-0782, Document
Referenced by:: §4.48(iv).
Encodings:: BibTeX

…

27: 28.4 Fourier Series

… ►

§28.4(vi) Behavior for Small $q$

… ►

§28.4(vii) Asymptotic Forms for Large $m$

… ►

28.4.25

\frac{A^{2n+1}_{2m+1}(q)}{A^{2n+1}_{1}(q)}=\frac{(-1)^{m+1}}{\left({\left(% \tfrac{1}{2}\right)_{m+1}}\right)^{2}}\left(\frac{q}{4}\right)^{m+1}\frac{2% \left(1+O\left(m^{-1}\right)\right)}{w^{\prime}_{\mbox{\tiny II}}(\frac{1}{2}% \pi;a_{2n+1}\left(q\right),q)},

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $w(z)$ : Mathieu’s equation solution and $A_{m}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.4.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §28.4(vii), §28.4 and Ch.28

… ►

28.4.27

\frac{B^{2n+2}_{2m}(q)}{B^{2n+2}_{2}(q)}=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{% q}{4}\right)^{m}\frac{q\pi\left(1+O\left(m^{-1}\right)\right)}{w^{\prime}_{% \mbox{\tiny I}}(\frac{1}{2}\pi;b_{2n+2}\left(q\right),q)}.

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $w(z)$ : Mathieu’s equation solution and $B_{m}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.4.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §28.4(vii), §28.4 and Ch.28

►For the basic solutions

w_{\mbox{\tiny I}}

and

w_{\mbox{\tiny II}}

see §28.2(ii).

28: 36.5 Stokes Sets

… ►The Stokes set takes different forms for

z=0

z<0

, and

z>0

. … ►where

u

satisfies the equation … ►The first sheet corresponds to

x<0

and is generated as a solution of Equations (36.5.6)–(36.5.9). …For

\left|Y\right|>Y_{1}

the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for

\left|Y\right|<Y_{1}

it is generated by the roots of the polynomial equation … ►Here

u

is the root of the equation …

29: Bibliography D

… ►

A. Debosscher (1998) Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:: ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… ►

T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

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External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.26, §2.8(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

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External Links:: ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt
Referenced by:: §2.8(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.8(vi), §8.12, §8.12.
Encodings:: BibTeX

… ►

T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

…

30: Bibliography R

… ►

W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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External Links:: MathReview (E. A. Boyd), ZentralBlatt
Referenced by:: (9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).
Encodings:: BibTeX

►

W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

►

W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

È. Ya. Riekstynš (1991) Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:: ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt
Referenced by:: §12.11(ii), §6.13.
Encodings:: BibTeX

… ►

G. M. Roper (1951) Some Applications of the Lamé Function Solutions of the Linearised Supersonic Flow Equations. Technical Reports and Memoranda Technical Report 2865, Aeronautical Research Council (Great Britain).

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Notes:: Reprinted by Her Majesty’s Stationery Office, London, 1962.
External Links:: MathReview (P. Germain)
Referenced by:: §29.19(ii).
Encodings:: BibTeX

…