Liouvillean solutions

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11: 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 ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. These solutions are called path-multiplicative. …

12: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

13: 32.1 Special Notation

… ►The functions treated in this chapter are the solutions of the Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$.

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

15: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

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§3.6(ii) Homogeneous Equations

… ► … ► … ►

§3.6(iv) Inhomogeneous Equations

…

16: 12.4 Power-Series Expansions

… ►

12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$U(a,z)$: 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:

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See also:

Annotations for §12.4 and Ch.12

►

12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$V(a,z)$: 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:

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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 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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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:

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Annotations for §12.4 and Ch.12

►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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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:

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See also:

Annotations for §12.4 and Ch.12

…

17: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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

Odd Second Solutions

… ►

Even Second Solutions

…

18: 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 $\mathrm{\Im }z\ge 0$ 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. …

19: 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).

20: 2.9 Difference Equations

§2.9 Difference Equations

… ►Formal solutions are … ►

§2.9(ii) Coincident Characteristic Values

… ►But there is an independent solution … ►