solutions near irregular singularities

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11: Bibliography

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A. S. Abdullaev (1985) Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:: English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47
External Links:: ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:: MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §23.21(ii).
Encodings:: BibTeX

►

H. Airault (1979) Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.

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External Links:: ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §32.10(ii), §32.8(i), §32.8(v).
Encodings:: BibTeX

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V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

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Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

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F. M. Arscott (1956) Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.

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External Links:: ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

…

12: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►These expansions are uniform with respect to

z

, including the turning point

z_{0}

and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►Then for large

u

asymptotic approximations of the solutions

w

can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on

u

and

\alpha

). …

13: 13.2 Definitions and Basic Properties

… ►This equation has a regular singularity at the origin with indices

0

and

1-b

, and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at

b

and

\infty

coalesce into an irregular singularity at

\infty

. ►

Standard Solutions

… ►

§13.2(v) Numerically Satisfactory Solutions

… ►A fundamental pair of solutions that is numerically satisfactory near the origin is …

14: 16.21 Differential Equation

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16.21.1

\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,

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Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $w$ : solution and $\vartheta$ : differential operator
Referenced by:: §16.21, §16.21
Permalink:: http://dlmf.nist.gov/16.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.21 and Ch.16

… ►With the classification of §16.8(i), when

p<q

the only singularities of (16.21.1) are a regular singularity at

z=0

and an irregular singularity at

z=\infty

. When

p=q

the only singularities of (16.21.1) are regular singularities at

z=0

(-1)^{p-m-n}

, and

\infty

. ►A fundamental set of solutions of (16.21.1) is given by …

15: Mark J. Ablowitz

… ►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations. …for appropriate data they can be linearized by the Inverse Scattering Transform (IST) and they possess solitons as special solutions. Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs which do not have moveable branch point singularities. ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …

16: 30.2 Differential Equations

… ►

30.2.1

\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{\mu^{2}}{1-z^{2}}\right)w=0.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter and $\mu$ : real parameter
Referenced by:: §30.11(i), §30.12, §30.12, §30.13(iv), §30.14(iv), §30.16(ii), §30.2(ii), §30.2(iii), §30.3(i), §30.4(i), §30.5, §30.6, §30.6
Permalink:: http://dlmf.nist.gov/30.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §30.2(i), §30.2 and Ch.30

►This equation has regular singularities at

z=\pm 1

with exponents

\pm\frac{1}{2}\mu

and an irregular singularity of rank 1 at

z=\infty

(if

\gamma\neq 0

). … … ►

30.2.4

(\zeta^{2}-\gamma^{2})\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+2\zeta% \frac{\mathrm{d}w}{\mathrm{d}\zeta}+\left(\zeta^{2}-\lambda-\gamma^{2}-\frac{% \gamma^{2}\mu^{2}}{\zeta^{2}-\gamma^{2}}\right)w=0.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $w$ : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter, $\mu$ : real parameter and $\zeta$ : parameter
Referenced by:: §30.2(iii)
Permalink:: http://dlmf.nist.gov/30.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §30.2(ii), §30.2 and Ch.30

…

17: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the

z

-plane that encircles

s_{1}

and

s_{2}

once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor

{\mathrm{e}}^{2\nu\pi i}

. These solutions are called path-multiplicative. …

18: 2.9 Difference Equations

§2.9 Difference Equations

… ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). Formal solutions are … ►But there is an independent solution … ►

19: 1.13 Differential Equations

… ►

Fundamental Pair

… ► … ►

§1.13(v) Products of Solutions

… ►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. ►

§1.13(vii) Closed-Form Solutions

…

20: 33.23 Methods of Computation

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§33.23(ii) Series Solutions

►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►This implies decreasing

\ell

for the regular solutions and increasing

\ell

for the irregular solutions of §§33.2(iii) and 33.14(iii). …