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
§13.7(ii) Error Bounds
§13.7(iii) Exponentially-Improved Expansion

§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 ${\mathbf{M}}\left(a,b,x\right)\sim\frac{e^{x}x^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}x^{-% s},$

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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 ), : nonnegative integer and : 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 ${\mathbf{M}}\left(a,b,z\right)\sim\frac{e^{z}z^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}z^{-% s}+\frac{e^{\pm\pi\mathrm{i}a}z^{-a}}{\Gamma\left(b-a\right)}\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$ $-\frac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\frac{3}{2}\pi-\delta$ ,

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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 ), $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, : nonnegative integer, : 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 $U\left(a,b,z\right)\sim z^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ .

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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 ), $\operatorname{ph}$ : phase, : nonnegative integer, : 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

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

. Magnify

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $\overline{\NVar{z}}$ : complex conjugate
Referenced by:: §13.7(ii)
Permalink:: http://dlmf.nist.gov/13.7.F1
Encodings:: pdf, png
See also:: Annotations for §13.7(ii), §13.7 and Ch.13

13.7.4 $U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+\varepsilon_{n}(z),$

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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 ), : nonnegative integer, : nonnegative integer, : 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 $\left|\varepsilon_{n}(z)\right|,~\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)% \right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right|\exp\left(\frac{2\alpha\rho C_{1}}{|z|}\right),$

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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 ), : nonnegative integer, : 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 $C_{n}=1,\quad\chi(n),\quad\left(\chi(n)+\sigma\nu^{2}n\right)\nu^{n},$

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Defines:: $C_{n}$ : coefficient (locally)
Symbols:: : 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 $z\in\textbf{R}_{1},\quad z\in\textbf{R}_{2}\cup\overline{\textbf{R}}_{2},\quad z% \in\textbf{R}_{3}\cup\overline{\textbf{R}}_{3},$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\in$ : element of, $\cup$ : union and : 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

$\sigma=\left|\ifrac{(b-2a)}{z}\right|$ ,

$\nu=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-4\sigma^{2}}\right)^{-\ifrac{1}{2}}$ ,

$\chi(n)=\sqrt{\pi}\Gamma\left(\tfrac{1}{2}n+1\right)/\Gamma\left(\tfrac{1}{2}n% +\tfrac{1}{2}\right)$ .

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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, : nonnegative integer and : 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

$\alpha=\frac{1}{1-\sigma}$ ,

$\beta=\frac{1-\sigma^{2}+\sigma|z|^{-1}}{2(1-\sigma)}$ ,

$\rho=\tfrac{1}{2}\left|2a^{2}-2ab+b\right|+\frac{\sigma(1+\frac{1}{4}\sigma)}{% (1-\sigma)^{2}}$ ,

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Defines:: $\alpha$ (locally), $\beta$ (locally) and $\rho$ (locally)
Symbols:: : 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 $U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+R_{n}(a,b,z),$

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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 ), : nonnegative integer, : nonnegative integer, : 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 $R_{n}(a,b,z)=\frac{(-1)^{n}2\pi z^{a-b}}{\Gamma\left(a\right)\Gamma\left(a-b+1% \right)}\left(\sum_{s=0}^{m-1}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s% }}}{s!}(-z)^{-s}G_{n+2a-b-s}(z)+{\left(1-a\right)_{m}}{\left(b-a\right)_{m}}R_% {m,n}(a,b,z)\right),$

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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 ), : integer, : nonnegative integer, : nonnegative integer, : 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 is an arbitrary nonnegative integer, and

13.7.12 $G_{p}(z)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z\right).$

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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 : 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 fixed

13.7.13 $R_{m,n}(a,b,z)=\begin{cases}O\left(e^{-|z|}z^{-m}\right),&|\operatorname{ph}z|% \leq\pi,\\ O\left(e^{z}z^{-m}\right),&\pi\leq|\operatorname{ph}z|\leq\tfrac{5}{2}\pi-% \delta.\\ \end{cases}$

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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, : integer, : nonnegative integer, : 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