13 Confluent Hypergeometric Functions Computation 13.30 Tables 13.32 Software

§13.31 Approximations

ⓘ

Keywords:: Kummer functions, approximations
Permalink:: http://dlmf.nist.gov/13.31
See also:: Annotations for Ch.13

§13.31(i) Chebyshev-Series Expansions
§13.31(ii) Padé Approximations
§13.31(iii) Rational Approximations

§13.31(i) Chebyshev-Series Expansions

ⓘ

Keywords:: Chebyshev-series expansions, Kummer functions
Permalink:: http://dlmf.nist.gov/13.31.i
See also:: Annotations for §13.31 and Ch.13

Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ that include the intervals $0\leq x\leq\alpha$ and $\alpha\leq x<\infty$ , respectively, where $\alpha$ is an arbitrary positive constant.

§13.31(ii) Padé Approximations

ⓘ

Permalink:: http://dlmf.nist.gov/13.31.ii
See also:: Annotations for §13.31 and Ch.13

For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985).

§13.31(iii) Rational Approximations

ⓘ

Keywords:: Kummer functions, computation
Permalink:: http://dlmf.nist.gov/13.31.iii
See also:: Annotations for §13.31 and Ch.13

In Luke (1977a) the following rational approximation is given, together with its rate of convergence. For the notation see §16.2(i).

Let $a,a+1-b\neq 0,-1,-2,\dots$ , $|\operatorname{ph}z|<\pi$ ,

13.31.1		$A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),$
ⓘ Defines: $A_{n}(z)$ : expansion (locally) Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.31.E1 Encodings: TeX, pMML, png See also: Annotations for §13.31(iii), §13.31 and Ch.13

and

13.31.2		$B_{n}(z)={{}_{2}F_{2}}\left({-n,n+1\atop a+1,b+1};-z\right).$
ⓘ Defines: $B_{n}(z)$ : expansion (locally) Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.31.E2 Encodings: TeX, pMML, png See also: Annotations for §13.31(iii), §13.31 and Ch.13

Then

13.31.3		$z^{a}U\left(a,1+a-b,z\right)=\lim_{n\to\infty}\frac{A_{n}(z)}{B_{n}(z)}.$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $n$ : nonnegative integer, $z$ : complex variable, $A_{n}(z)$ : expansion and $B_{n}(z)$ : expansion Permalink: http://dlmf.nist.gov/13.31.E3 Encodings: TeX, pMML, png See also: Annotations for §13.31(iii), §13.31 and Ch.13

13.30 Tables 13.32 Software

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§13.31 Approximations

Contents

§13.31(i) Chebyshev-Series Expansions

§13.31(ii) Padé Approximations

§13.31(iii) Rational Approximations