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11: 32.1 Special Notation

… ►The functions treated in this chapter are the solutions of the Painlevé equations

\mbox{P}_{\mbox{\scriptsize I}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

… ►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 with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …

13: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

… ►

§3.6(ii) Homogeneous Equations

… ► … ► … ►

§3.6(iv) Inhomogeneous Equations

…

14: 12.4 Power-Series Expansions

… ►

12.4.1

U\left(a,z\right)=U\left(a,0\right)u_{1}(a,z)+U'\left(a,0\right)u_{2}(a,z),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

►

12.4.2

V\left(a,z\right)=V\left(a,0\right)u_{1}(a,z)+V'\left(a,0\right)u_{2}(a,z),

ⓘ

Symbols:: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

►where the initial values are given by (12.2.6)–(12.2.9), and

u_{1}(a,z)

and

u_{2}(a,z)

are the even and odd solutions of (12.2.2) given by ►

12.4.3

u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),

ⓘ

Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.1 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

►

12.4.4

u_{2}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(z+(a+\tfrac{3}{2})\frac{z^{3}}{3!}+(a+% \tfrac{3}{2})(a+\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).

ⓘ

Defines:: $u_{2}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.3 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

…

15: 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

… ►Second solutions of (28.2.1) are given by … … ► … ►

Odd Second Solutions

… ►

Even Second Solutions

…

16: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►The quotient of two solutions of (15.10.1) maps the closed upper half-plane

\Im z\geq 0

conformally onto a curvilinear triangle. … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

17: 14.29 Generalizations

… ►Solutions of the equation …As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when

\mu_{1}=\mu_{2}=\mu

. … ►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).

18: 2.9 Difference Equations

§2.9 Difference Equations

… ►Formal solutions are … ►

§2.9(ii) Coincident Characteristic Values

… ►But there is an independent solution … ►

19: 28.34 Methods of Computation

… ►

(b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$ ; see Schäfke (1961a).

… ►

(d)

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

►

(e)

Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

… ►

§28.34(iii) Floquet Solutions

… ►

(c)

Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

…

20: 31.8 Solutions via Quadratures

§31.8 Solutions via Quadratures

… ►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … ►are two independent solutions of (31.2.1). … ►For more details see Smirnov (2002). ►The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).