§13.8 Asymptotic Approximations for Large Parameters

If $b\to \mathrm{\infty }$ in $ℂ$ in such a way that $\left|b+n\right|\ge \delta >0$ for all $n=0,1,2,\mathrm{\dots }$, then

13.8.1		$$M(a,b,z)=\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s+O\left({|b|}^{-n}\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.9(ii) Permalink: http://dlmf.nist.gov/13.8.E1 Encodings: TeX, pMML, png See also: Annotations for §13.8(i), §13.8 and Ch.13

For fixed $a$ and $z$ in $ℂ$

13.8.2		$$M(a,b,z)\sim \frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}{\left(a\right)}_s{q}_s(z,a)b^{-s-a},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.8(i) Permalink: http://dlmf.nist.gov/13.8.E2 Encodings: TeX, pMML, png See also: Annotations for §13.8(i), §13.8 and Ch.13

as $b\to \mathrm{\infty }$ in $|\mathrm{ph}b|\le \pi -\delta$, where $q_0(z,a)=1$ and

13.8.3		$${\left({\mathrm{e}}^t-1\right)}^{a-1}\exp \left(t+z(1-{\mathrm{e}}^{-t})\right)=\sum _{s=0}^{\mathrm{\infty }}{q}_s(z,a)t^{s+a-1}.$$
ⓘ Symbols: $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.8(i) Permalink: http://dlmf.nist.gov/13.8.E3 Encodings: TeX, pMML, png See also: Annotations for §13.8(i), §13.8 and Ch.13

When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded.

Let $\lambda =z/b>0$ and $\zeta =\sqrt{2(\lambda -1-\ln \lambda )}$ with $\mathrm{sign}\left(\zeta \right)=\mathrm{sign}\left(\lambda -1\right)$. Then

13.8.4		$$M(a,b,z)\sim b^{\frac{1}{2}a}{\mathrm{e}}^{\frac{1}{4}{\zeta }^{2}b}(\begin{array}{l}\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}U(a-\frac{1}{2},-\zeta \sqrt{b})\\ +\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}-{\left(\frac{\zeta }{\lambda -1}\right)}^a\right)\frac{U(a-\frac{3}{2},-\zeta \sqrt{b})}{\zeta \sqrt{b}})\end{array}$$
ⓘ Symbols: $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.8.E4 Encodings: TeX, pMML, png See also: Annotations for §13.8(ii), §13.8 and Ch.13

and

13.8.5		$$U(a,b,z)\sim b^{-\frac{1}{2}a}{\mathrm{e}}^{\frac{1}{4}{\zeta }^{2}b}(\begin{array}{l}\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}U(a-\frac{1}{2},\zeta \sqrt{b})\\ -\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}-{\left(\frac{\zeta }{\lambda -1}\right)}^a\right)\frac{U(a-\frac{3}{2},\zeta \sqrt{b})}{\zeta \sqrt{b}})\end{array}$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.8.E5 Encodings: TeX, pMML, png See also: Annotations for §13.8(ii), §13.8 and Ch.13

as $b\to \mathrm{\infty }$, uniformly in compact $\lambda$-intervals of $(0,\mathrm{\infty })$ and compact real $a$-intervals. For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

Special cases are

13.8.6		$$M(a,b,b)=\sqrt{\pi }{\left(\frac{b}{2}\right)}^{\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/13.8.E6 Encodings: TeX, pMML, png See also: Annotations for §13.8(ii), §13.8 and Ch.13

and

13.8.7		$$U(a,b,b)=\sqrt{\pi }{\left(2b\right)}^{-\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/13.8.E7 Encodings: TeX, pMML, png See also: Annotations for §13.8(ii), §13.8 and Ch.13

To obtain approximations for $M(a,b,z)$ and $U(a,b,z)$ that hold as $b\to \mathrm{\infty }$, with $a>\frac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i).

Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv).

For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010).

For more asymptotic expansions for the cases $b\to \pm \mathrm{\infty }$ see Temme (2015, §§10.4 and 22.5)

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When $a\to +\mathrm{\infty }$ with $b$ ($\le 1$) fixed,

13.8.8		$$U(a,b,x)=\frac{2{\mathrm{e}}^{\frac{1}{2}x}}{\mathrm{\Gamma }\left(a\right)}(\begin{array}{l}\sqrt{\frac{2}{\beta }\tanh \left(\frac{w}{2}\right)}{\left(\frac{1-{\mathrm{e}}^{-w}}{\beta }\right)}^{-b}{\beta }^{1-b}{K}_{1-b}\left(2\beta a\right)\\ +a^{-1}{\left(\frac{a^{-1}+\beta }{1+\beta }\right)}^{1-b}{\mathrm{e}}^{-2\beta a}O\left(1\right)),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\tanh z$: hyperbolic tangent function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable Referenced by: §13.8(iii) Permalink: http://dlmf.nist.gov/13.8.E8 Encodings: TeX, pMML, png See also: Annotations for §13.8(iii), §13.8 and Ch.13

where $w=\mathrm{arccosh}\left(1+{(2a)}^{-1}x\right)$, and $\beta =(w+\sinh w)/2$. (13.8.8) holds uniformly with respect to $x\in [0,\mathrm{\infty })$. For the case $b>1$ the transformation (13.2.40) can be used.

For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).

When $a\to -\mathrm{\infty }$ with $b$ ($\ge 1$) fixed,

13.8.9		$$M(a,b,x)=\begin{array}{l}\mathrm{\Gamma }\left(b\right){\mathrm{e}}^{\frac{1}{2}x}{\left((\frac{1}{2}b-a)x\right)}^{\frac{1}{2}-\frac{1}{2}b}\\ \times (\begin{array}{l}{J}_{b-1}\left(\sqrt{2x(b-2a)}\right)\\ +\mathrm{env}{J}_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a\right|}^{-\frac{1}{2}}\right)),\end{array}\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm and $x$: real variable A&S Ref: 13.5.13 (in different form) Referenced by: §13.9(i) Permalink: http://dlmf.nist.gov/13.8.E9 Encodings: TeX, pMML, png See also: Annotations for §13.8(iii), §13.8 and Ch.13

and

13.8.10		$$U(a,b,x)=\begin{array}{l}\mathrm{\Gamma }\left(\frac{1}{2}b-a+\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}x}{x}^{\frac{1}{2}-\frac{1}{2}b}\\ \times (\begin{array}{l}\begin{array}{l}\cos \left(a\pi \right)J_{b-1}\left(\sqrt{2x(b-2a)}\right)\\ -\sin \left(a\pi \right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right)\end{array}\\ +\mathrm{env}{Y}_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a\right|}^{-\frac{1}{2}}\right)),\end{array}\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable A&S Ref: 13.5.15 (in different form) Referenced by: §13.9(ii) Permalink: http://dlmf.nist.gov/13.8.E10 Encodings: TeX, pMML, png See also: Annotations for §13.8(iii), §13.8 and Ch.13

uniformly with respect to bounded positive values of $x$ in each case.

For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).

When $a\to \mathrm{\infty }$ in $\left|\mathrm{ph}a\right|\le \pi -\delta$ and $b$ and $z$ fixed,

13.8.11		$$U(a,b,z)\sim 2{\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}}{\mathrm{\Gamma }\left(a\right)}(\begin{array}{l}{K}_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}\\ +\sqrt{z/a}{K}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}),\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.8(iii) Permalink: http://dlmf.nist.gov/13.8.E11 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. See also: Annotations for §13.8(iii), §13.8 and Ch.13

13.8.12		$$\mathbf{M}(a,b,z)\sim \begin{array}{l}{\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a-b\right)}{\mathrm{\Gamma }\left(a\right)}\\ \times (\begin{array}{l}{I}_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}\\ -\sqrt{z/a}{I}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}),\end{array}\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.8.E12 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. See also: Annotations for §13.8(iii), §13.8 and Ch.13

13.8.13		$$\mathbf{M}(-a,b,z)\sim \begin{array}{l}{\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)}{\mathrm{\Gamma }\left(a+b\right)}\\ \times (\begin{array}{l}{J}_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}\\ -\sqrt{z/a}{J}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}),\end{array}\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.13) Permalink: http://dlmf.nist.gov/13.8.E13 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.13) is (10.3.58) in Temme (2015). In that reference the ${(-1)}^s$ is missing. See also: Annotations for §13.8(iii), §13.8 and Ch.13

13.8.14		$$U(-a,b,z)\sim \begin{array}{l}{\left(z/a\right)}^{(1-b)/2}{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)\\ \times (\begin{array}{l}{C}_{b-1}(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}\\ -\sqrt{z/a}{C}_b(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}),\end{array}\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.14) Permalink: http://dlmf.nist.gov/13.8.E14 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.14) is (10.3.68) in Temme (2015). In that reference the ${(-1)}^s$ is missing. See also: Annotations for §13.8(iii), §13.8 and Ch.13

where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and

13.8.15		$p_k(z)$	$={\displaystyle \sum _{s=0}^k}\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{s}}\right){\left(1-b+s\right)}_{k-s}{z}^s{c}_{k+s}(z),$
		$q_k(z)$	$={\displaystyle \sum _{s=0}^k}\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{s}}\right){\left(2-b+s\right)}_{k-s}{z}^s{c}_{k+s+1}(z)$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.15) Permalink: http://dlmf.nist.gov/13.8.E15 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.15) is (10.3.41) in Temme (2015). In that reference there are typographical errors in the Pochhammer symbols. See also: Annotations for §13.8(iii), §13.8 and Ch.13

where $c_0(z)=1$ and

13.8.16		$$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$
$k=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.16), §13.8(iii) Permalink: http://dlmf.nist.gov/13.8.E16 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies, $$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$ For (13.8.16) combine this differential equation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1). See also: Annotations for §13.8(iii), §13.8 and Ch.13

For the Bernoulli numbers $B_k$ see §24.2(i) and for proofs and similar results in which $z$ can also be unbounded see Temme (2015, Chapters 10 and 27)