Olver confluent hypergeometric function

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11: 13 Confluent Hypergeometric Functions

Chapter 13 Confluent Hypergeometric Functions

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12: 13.29 Methods of Computation

… ►For $M(a,b,z)$ and $M_{\kappa ,\mu }\left(z\right)$ this means that in the sector $|\mathrm{ph}z|\le \pi$ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. In the following two examples Olver’s algorithm (§3.6(v)) can be used. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of $M(n,b,x)$, when $b$ and $x$ are real and $n$ is a positive integer. …

13: Bibliography T

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Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

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External Links:

ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

BibTeX

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14: 13.22 Zeros

§13.22 Zeros

►From (13.14.2) and (13.14.3) $M_{\kappa ,\mu }\left(z\right)$ has the same zeros as $M(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ and $W_{\kappa ,\mu }\left(z\right)$ has the same zeros as $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$, hence the results given in §13.9 can be adopted. … ►For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by …where $j_{2\mu ,r}$ is the $r$th positive zero of the Bessel function $J_{2\mu }\left(x\right)$ (§10.21(i)). …

15: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4). … ►These results are proved in Olver (1980b). … ►These results are proved in Olver (1980b). …Olver (1980b) also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively. … ►For uniform approximations of $M_{\kappa ,\mathrm{i}\mu }\left(z\right)$ and $W_{\kappa ,\mathrm{i}\mu }\left(z\right)$, $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a). …

16: 13.16 Integral Representations

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13.16.9 $$W_{\kappa ,\mu }\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa +c}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{c-1}{}_2{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}-\mu -\kappa }{c};-t)dt,$$ ,

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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17: 13.14 Definitions and Basic Properties

… ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu }$ and $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. …

18: 10.16 Relations to Other Functions

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Elementary Functions

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Confluent Hypergeometric Functions

… ►For the functions $M$ and $U$ see §13.2(i). …For the functions $M_{0,\nu }$ and $W_{0,\nu }$ see §13.14(i). … ►

Generalized Hypergeometric Functions

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19: 16.25 Methods of Computation

§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. …See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).

20: Bibliography D

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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