13 Confluent Hypergeometric Functions Kummer Functions 13.6 Relations to Other Functions 13.8 Asymptotic Approximations for Large Parameters

§13.7 Asymptotic Expansions for Large Argument

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Keywords:: Kummer functions, asymptotic expansions for large argument
Referenced by:: §13.29(i), §13.29(ii)
Permalink:: http://dlmf.nist.gov/13.7
See also:: Annotations for Ch.13

§13.7(i) Poincaré-Type Expansions

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Notes:: See Temme (1996b, §7.2) or Olver (1997b, pp. 256–258).
Permalink:: http://dlmf.nist.gov/13.7.i
See also:: Annotations for §13.7 and Ch.13

As $x\to\infty$

13.7.1		${\mathbf{M}}\left(a,b,x\right)\sim\frac{e^{x}x^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}x^{-% s},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/13.7.E1 Encodings: TeX, pMML, png See also: Annotations for §13.7(i), §13.7 and Ch.13

provided that $a\neq 0,-1,\dots$ .

As $z\to\infty$

13.7.2		${\mathbf{M}}\left(a,b,z\right)\sim\frac{e^{z}z^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}z^{-% s}+\frac{e^{\pm\pi\mathrm{i}a}z^{-a}}{\Gamma\left(b-a\right)}\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$
$-\frac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\frac{3}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : 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, $\operatorname{ph}$ : phase, $s$ : nonnegative integer, $z$ : complex variable and $\delta$ : small positive constant A&S Ref: 13.5.1 (with corrected region of validity) Referenced by: §13.2(iv), §13.7(ii), §13.9(i) Permalink: http://dlmf.nist.gov/13.7.E2 Encodings: TeX, pMML, png See also: Annotations for §13.7(i), §13.7 and Ch.13

unless $a=0,-1,\dots$ and $b-a=0,-1,\dots$ . Here $\delta$ denotes an arbitrary small positive constant. Also,

13.7.3		$U\left(a,b,z\right)\sim z^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$
$\|\operatorname{ph}z\|\leq\tfrac{3}{2}\pi-\delta$ .
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : phase, $s$ : nonnegative integer, $z$ : complex variable and $\delta$ : small positive constant A&S Ref: 13.5.2 Referenced by: §12.9(i), §13.12, §13.2(i) Permalink: http://dlmf.nist.gov/13.7.E3 Encodings: TeX, pMML, png See also: Annotations for §13.7(i), §13.7 and Ch.13

§13.7(ii) Error Bounds

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Keywords:: Kummer functions, asymptotic expansions for large argument, error bounds
Notes:: See Olver (1965).
Referenced by:: §12.9(ii), §13.19, Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/13.7.ii
See also:: Annotations for §13.7 and Ch.13

See accompanying text — Figure 13.7.1: Regions ${\textbf{R}}_{1}$ , ${\textbf{R}}_{2}$ , $\overline{\textbf{R}}_{2}$ , ${\textbf{R}}_{3}$ , and $\overline{\textbf{R}}_{3}$ are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with $r=|b-2a|$ . Magnify

13.7.4		$U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+\varepsilon_{n}(z),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer, $z$ : complex variable and $\varepsilon_{n}(z)$ : function Referenced by: §13.2(i), §13.7(ii) Permalink: http://dlmf.nist.gov/13.7.E4 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

where

13.7.5		$\left\|\varepsilon_{n}(z)\right\|,~\beta^{-1}\left\|\varepsilon_{n}^{\prime}(z)% \right\|\leq 2\alpha C_{n}\left\|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right\|\exp\left(\frac{2\alpha\rho C_{1}}{\|z\|}\right),$
ⓘ Defines: $\varepsilon_{n}(z)$ : function (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\exp\NVar{z}$ : exponential function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $z$ : complex variable, $C_{n}$ : coefficient, $\alpha$ , $\beta$ and $\rho$ Referenced by: §13.2(i) Permalink: http://dlmf.nist.gov/13.7.E5 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

and with the notation of Figure 13.7.1

13.7.6		$C_{n}=1,\quad\chi(n),\quad\left(\chi(n)+\sigma\nu^{2}n\right)\nu^{n},$
ⓘ Defines: $C_{n}$ : coefficient (locally) Symbols: $n$ : nonnegative integer, $\sigma$ , $\nu$ and $\chi(n)$ : function Permalink: http://dlmf.nist.gov/13.7.E6 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

according as

13.7.7		$z\in\textbf{R}_{1},\quad z\in\textbf{R}_{2}\cup\overline{\textbf{R}}_{2},\quad z% \in\textbf{R}_{3}\cup\overline{\textbf{R}}_{3},$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $\in$ : element of, $\cup$ : union and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.7.E7 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

respectively, with

13.7.8	$\displaystyle\sigma$	$\displaystyle=\left\|\ifrac{(b-2a)}{z}\right\|$ ,
	$\displaystyle\nu$	$\displaystyle=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-4\sigma^{2}}\right)^{-% \ifrac{1}{2}}$ ,
	$\displaystyle\chi(n)$	$\displaystyle=\sqrt{\pi}\Gamma\left(\tfrac{1}{2}n+1\right)/\Gamma\left(\tfrac{% 1}{2}n+\tfrac{1}{2}\right)$ .
ⓘ Defines: $\sigma$ (locally), $\nu$ (locally) and $\chi(n)$ : function (locally) Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.7.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

Also, when $z\in\textbf{R}_{1}\cup\textbf{R}_{2}\cup\overline{\textbf{R}}_{2}$

13.7.9	$\displaystyle\alpha$	$\displaystyle=\frac{1}{1-\sigma}$ ,
	$\displaystyle\beta$	$\displaystyle=\frac{1-\sigma^{2}+\sigma\|z\|^{-1}}{2(1-\sigma)}$ ,
	$\displaystyle\rho$	$\displaystyle=\tfrac{1}{2}\left\|2a^{2}-2ab+b\right\|+\frac{\sigma(1+\frac{1}{4}% \sigma)}{(1-\sigma)^{2}}$ ,
ⓘ Defines: $\alpha$ (locally), $\beta$ (locally) and $\rho$ (locally) Symbols: $z$ : complex variable and $\sigma$ Referenced by: §13.7(ii), §13.7(ii) Permalink: http://dlmf.nist.gov/13.7.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

and when $z\in\textbf{R}_{3}\cup\overline{\textbf{R}}_{3}$ $\sigma$ is replaced by $\nu\sigma$ and $|z|^{-1}$ is replaced by $\nu|z|^{-1}$ everywhere in (13.7.9).

For numerical values of $\chi(n)$ see Table 9.7.1.

Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9).

§13.7(iii) Exponentially-Improved Expansion

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Keywords:: Kummer functions, asymptotic expansions for large argument, exponentially-improved, hyperasymptotic
Referenced by:: §12.9(ii), §13.19, Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/13.7.iii
Addition (effective with 1.0.10):: A sentence and reference to Paris (2013) were added at the end of this subsection.
See also:: Annotations for §13.7 and Ch.13

Let

13.7.10		$U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+R_{n}(a,b,z),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer, $z$ : complex variable and $R_{n}(a,b,z)$ : remainder Referenced by: §13.29(i) Permalink: http://dlmf.nist.gov/13.7.E10 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

and

13.7.11		$R_{n}(a,b,z)=\frac{(-1)^{n}2\pi z^{a-b}}{\Gamma\left(a\right)\Gamma\left(a-b+1% \right)}\left(\sum_{s=0}^{m-1}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s% }}}{s!}(-z)^{-s}G_{n+2a-b-s}(z)+{\left(1-a\right)_{m}}{\left(b-a\right)_{m}}R_% {m,n}(a,b,z)\right),$
ⓘ Defines: $R_{n}(a,b,z)$ : remainder (locally) Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer, $n$ : nonnegative integer, $s$ : nonnegative integer, $z$ : complex variable and $G_{p}(z)$ : expansion Referenced by: §13.29(i) Permalink: http://dlmf.nist.gov/13.7.E11 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

where $m$ is an arbitrary nonnegative integer, and

13.7.12		$G_{p}(z)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z\right).$
ⓘ Defines: $G_{p}(z)$ : expansion (locally) Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.7.E12 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

(For the notation see §8.2(i).) Then as $z\to\infty$ with $\left|\left|z\right|-n\right|$ bounded and $a, b, m$ fixed

13.7.13		$R_{m,n}(a,b,z)=\begin{cases}O\left(e^{-\|z\|}z^{-m}\right),&\|\operatorname{ph}z\|% \leq\pi,\\ O\left(e^{z}z^{-m}\right),&\pi\leq\|\operatorname{ph}z\|\leq\tfrac{5}{2}\pi-% \delta.\\ \end{cases}$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $m$ : integer, $n$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant and $R_{n}(a,b,z)$ : remainder Permalink: http://dlmf.nist.gov/13.7.E13 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

For proofs see Olver (1991b, 1993a). For the special case $\operatorname{ph}z=\pm\pi$ see Paris (2013). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

13.6 Relations to Other Functions 13.8 Asymptotic Approximations for Large Parameters

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§13.7 Asymptotic Expansions for Large Argument

Contents

§13.7(i) Poincaré-Type Expansions

§13.7(ii) Error Bounds

§13.7(iii) Exponentially-Improved Expansion