Poincaré type

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11: 9.12 Scorer Functions

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Mellin–Barnes Type Integral

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9.12.25 $$\mathrm{Gi}\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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9.12.26 $${\mathrm{Gi}}^{\prime }\left(z\right)\sim -\frac{1}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k+1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Refer to §2.1(ii).

Permalink:

http://dlmf.nist.gov/9.12.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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9.12.27 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.31)

Permalink:

http://dlmf.nist.gov/9.12.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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9.12.28 $${\mathrm{Hi}}^{\prime }\left(z\right)\sim \frac{1}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k+1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$.

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Refer to §2.1(ii).

Permalink:

http://dlmf.nist.gov/9.12.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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12: 2.7 Differential Equations

… ►The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►

2.7.12 $$a_{s,1}\sim \frac{{\mathrm{\Lambda }}_1}{{({\lambda }_1-{\lambda }_2)}^s}\sum _{j=0}^{\mathrm{\infty }}{a}_{j,2}{({\lambda }_1-{\lambda }_2)}^j\mathrm{\Gamma }\left(s+{\mu }_2-{\mu }_1-j\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and ${\mathrm{\Lambda }}_j$: constants

Referenced by:

§2.11(v), §2.7(ii)

Permalink:

http://dlmf.nist.gov/2.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(ii), §2.7 and Ch.2

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2.7.13 $$a_{s,2}\sim \frac{{\mathrm{\Lambda }}_2}{{({\lambda }_2-{\lambda }_1)}^s}\sum _{j=0}^{\mathrm{\infty }}{a}_{j,1}{({\lambda }_2-{\lambda }_1)}^j\mathrm{\Gamma }\left(s+{\mu }_1-{\mu }_2-j\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and ${\mathrm{\Lambda }}_j$: constants

Referenced by:

§2.11(v), §2.7(ii)

Permalink:

http://dlmf.nist.gov/2.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(ii), §2.7 and Ch.2

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2.7.14 $$w_j(z)\sim {\mathrm{e}}^{{\lambda }_{j}z}{(({\lambda }_2-{\lambda }_1)z)}^{{\mu }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{z^s}$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and $w_j(z)$: solutions

Referenced by:

§2.11(v), §2.7(ii), §2.7(ii)

Permalink:

http://dlmf.nist.gov/2.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(ii), §2.7 and Ch.2

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13: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

Encodings:

BibTeX

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Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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