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11: 10.57 Uniform Asymptotic Expansions for Large Order

… ►Subsequently, for ${𝗂}_n^{(2)}\left((n+\frac{1}{2})z\right)$ the connection formula (10.47.11) is available. …

12: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

13: 19.7 Connection Formulas

§19.7 Connection Formulas

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14: 12.2 Differential Equations

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§12.2(v) Connection Formulas

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15: Bibliography I

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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

BibTeX

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16: 31.12 Confluent Forms of Heun’s Equation

… ►For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …

17: 32.11 Asymptotic Approximations for Real Variables

… ►Connection formulas for $d$ and ${\theta }_0$ are given by … ►The connection formulas for $k$ are … ►The connection formulas for $\sigma$, $\rho$, and $\theta$ are … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Connection formulas for $d$ and ${\theta }_0$ are given by …

18: 16.8 Differential Equations

… ►We have the connection formula …

19: 25.12 Polylogarithms

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25.12.3 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{z}{z-1}\right)=-\frac{1}{2}{(\ln \left(1-z\right))}^2,$$ $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

connectionformula

Source:

Maximon (2003, (3.4), p. 2808)

A&S Ref:

27.7.5 (is a modified form with $z=1-x$)

Permalink:

http://dlmf.nist.gov/25.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.4 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{1}{z}\right)=-\frac{1}{6}{\pi }^2-\frac{1}{2}{(\ln \left(-z\right))}^2,$$ $z\in ℂ\setminus [0,\mathrm{\infty })$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction and $z$: complex variable

Keywords:

connectionformula

Source:

Maximon (2003, (3.2), p. 2808)

Referenced by:

§25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.6 $${\mathrm{Li}}_2\left(x\right)+{\mathrm{Li}}_2\left(1-x\right)=\frac{1}{6}{\pi }^2-(\ln x)\ln \left(1-x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\ln z$: principal branch of logarithm function and $x$: real variable

Keywords:

connectionformula

Source:

Maximon (2003, (3.3), p. 2808)

Permalink:

http://dlmf.nist.gov/25.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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20: Bibliography W

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On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. (Singap.) 7 (4), pp. 419–448.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(v), §32.11(v), §32.13(ii).

Encodings:

BibTeX

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On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

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External Links:

ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt

Referenced by:

§32.13(ii).

Encodings:

BibTeX

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