properties of solutions

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21: 28.12 Definitions and Basic Properties

… ►They have the following pseudoperiodic and orthogonality properties: …

22: 9.13 Generalized Airy Functions

… ►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 …

23: 4.37 Inverse Hyperbolic Functions

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4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

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Defines:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

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§4.37(v) Fundamental Property

►With $k\in \mathbb{Z}$, the general solutions of the equations …

24: 4.23 Inverse Trigonometric Functions

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4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

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Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

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§4.23(v) Fundamental Property

►With $k\in \mathbb{Z}$, the general solutions of the equations …

25: Bibliography F

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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

…

26: 9.12 Scorer Functions

… ►The general solution is given by …Standard particular solutions are … ►

§9.12(iv) Numerically Satisfactory Solutions

… ►In $ℂ$, numerically satisfactory sets of solutions are given by … ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

27: 9.16 Physical Applications

… ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …

28: 13.2 Definitions and Basic Properties

… ►Another standard solution of (13.2.1) is $U(a,b,z)$, which is determined uniquely by the property …

29: 10.47 Definitions and Basic Properties

§10.47 Definitions and Basic Properties

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§10.47(ii) Standard Solutions

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§10.47(iii) Numerically Satisfactory Pairs of Solutions

►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $𝗃$, $𝗒$, $𝗁$, and $n$, respectively. …

30: 31.8 Solutions via Quadratures

§31.8 Solutions via Quadratures

… ►are two independent solutions of (31.2.1). … ►The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …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).