§13.7 Asymptotic Expansions for Large Argument

As $x\to \mathrm{\infty }$

13.7.1		$$\mathbf{M}(a,b,x)\sim \frac{{\mathrm{e}}^x{x}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{x}^{-s},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: 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 $a\ne 0,-1,\mathrm{\dots }$.

As $z\to \mathrm{\infty }$

13.7.2		$$\mathbf{M}(a,b,z)\sim \frac{{\mathrm{e}}^z{z}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{z}^{-s}+\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}{z}^{-a}}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$
$-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: 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, $\mathrm{ph}$: 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 $a=0,-1,\mathrm{\dots }$ and $b-a=0,-1,\mathrm{\dots }$. Here $\delta$ denotes an arbitrary small positive constant. Also,

13.7.3		$$U(a,b,z)\sim z^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$
$|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: 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 $n!$), $\mathrm{ph}$: 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

13.7.4		$$U(a,b,z)=z^{-a}\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s}+{\epsilon }_n(z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and ${\epsilon }_n(z)$: 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		$$\left|{\epsilon }_n(z)\right|,{\beta }^{-1}\left|{\epsilon }_n^{\prime }(z)\right|\le 2\alpha C_n\left|\frac{{\left(a\right)}_n{\left(a-b+1\right)}_n}{n!z^{a+n}}\right|\exp \left(\frac{2\alpha \rho C_1}{|z|}\right),$$
ⓘ Defines: ${\epsilon }_n(z)$: function (locally) Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\exp z$: 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		$$C_n=1,\chi (n),\left(\chi (n)+\sigma {\nu }^{2}n\right){\nu }^n,$$
ⓘ 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		$$z\in {\mathbf{\text{R}}}_1,z\in {\mathbf{\text{R}}}_2\cup {\overline{\mathbf{\text{R}}}}_2,z\in {\mathbf{\text{R}}}_3\cup {\overline{\mathbf{\text{R}}}}_3,$$
ⓘ Symbols: $\overline{z}$: 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		$\sigma$	$=\left|(b-2a)/z\right|$,
		$\nu$	$={\left(\frac{1}{2}+\frac{1}{2}\sqrt{1-4{\sigma }^2}\right)}^{-1/2}$,
		$\chi (n)$	$=\sqrt{\pi }\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)$.
ⓘ Defines: $\sigma$ (locally), $\nu$ (locally) and $\chi (n)$: function (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 $z\in {\mathbf{\text{R}}}_1\cup {\mathbf{\text{R}}}_2\cup {\overline{\mathbf{\text{R}}}}_2$

13.7.9		$\alpha$	$={\displaystyle \frac{1}{1-\sigma }}$,
		$\beta$	$={\displaystyle \frac{1-{\sigma }^2+\sigma {|z|}^{-1}}{2(1-\sigma )}}$,
		$\rho$	$=\frac{1}{2}\left|2a^2-2ab+b\right|+{\displaystyle \frac{\sigma (1+\frac{1}{4}\sigma )}{{(1-\sigma )}^2}}$,
ⓘ 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 $z\in {\mathbf{\text{R}}}_3\cup {\overline{\mathbf{\text{R}}}}_3$ $\sigma$ 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).

Let

13.7.10		$$U(a,b,z)=z^{-a}\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s}+R_n(a,b,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: 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		$$R_n(a,b,z)=\frac{{(-1)}^{n}2\pi z^{a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}(\begin{array}{l}\sum _{s=0}^{m-1}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{(-z)}^{-s}{G}_{n+2a-b-s}(z)\\ +{\left(1-a\right)}_m{\left(b-a\right)}_m{R}_{m,n}(a,b,z)),\end{array}$$
ⓘ Defines: $R_n(a,b,z)$: remainder (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: 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		$$G_p(z)=\frac{{\mathrm{e}}^z}{2\pi }\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }(1-p,z).$$
ⓘ Defines: $G_p(z)$: expansion (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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 $z\to \mathrm{\infty }$ with $\left|\left|z\right|-n\right|$ bounded and $a,b,m$ fixed

13.7.13
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: 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 $\mathrm{ph}z=\pm \pi$ see Paris (2013). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).