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21: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

►provided that both $\mu \mp \kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$. … ►

13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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22: Bibliography K

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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.

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External Links:

Document, ZentralBlatt

Referenced by:

2nd item, §10.77(viii).

Encodings:

BibTeX

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The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.

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Notes:

For Matlab software for this function, see Plamen Koev’s home page.

External Links:

Home Page, ISSN 0025-5718, Document, MathReview (Luis Verde-Star), ZentralBlatt

Referenced by:

§35.10, 2nd item.

Encodings:

BibTeX

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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Hypergeometric functions of $2\times 2$ matrix argument are expressible in terms of Appel’s functions $F_4$ . Proc. Amer. Math. Soc. 70 (1), pp. 39–42.

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External Links:

FullText, MathReview (P. N. Rathie), ZentralBlatt

Referenced by:

§35.7(ii).

Encodings:

BibTeX

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23: 35 Functions of Matrix Argument

Chapter 35 Functions of Matrix Argument

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24: 8.28 Software

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§8.28(ii) Incomplete Gamma Functions for Real Argument and Parameter

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§8.28(iii) Incomplete Gamma Functions for Complex Argument and Parameter

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§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters

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§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters

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§8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter

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25: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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§10.40(ii) Error Bounds for Real Argument and Order

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§10.40(iii) Error Bounds for Complex Argument and Order

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26: Bibliography B

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New tables of Bessel functions of complex argument. Comput. Math. Math. Phys. 37 (12), pp. 1480–1482.

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Notes:

Russian original in Zh. Vychisl. Mat. i Mat. Fiz. 37(1997), no. 12, 1526–1528

External Links:

ZentralBlatt

Referenced by:

§10.75(i).

Encodings:

BibTeX

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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The behavior at unit argument of the hypergeometric function ${}_3{}^{}F_2^{}$ . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.

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External Links:

ISSN 0036-1410, Document, MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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External Links:

ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt

Referenced by:

§16.4(ii), §16.8(ii).

Encodings:

BibTeX

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Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.

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External Links:

ISSN 0377-0427, Document, MathReview (S. Bhargava), ZentralBlatt

Referenced by:

§35.5(ii), §35.5(iii).

Encodings:

BibTeX

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27: 20.16 Software

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§20.16(ii) Real Argument and Parameter

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§20.16(iii) Complex Argument and/or Parameter

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28: 22.22 Software

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§22.22(ii) Real Argument

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§22.22(iii) Complex Argument

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29: 15.20 Software

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§15.20(ii) Real Parameters and Argument

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§15.20(iii) Complex Parameters and Argument

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30: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(ii) Asymptotic Expansions of Derivatives

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(iv) Error Bounds for Complex Argument and Order

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§10.17(v) Exponentially-Improved Expansions

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