§13.8 Asymptotic Approximations for Large Parameters

Referenced by:: §13.29(i)
Permalink:: http://dlmf.nist.gov/13.8

§13.8(i) Large $|b|$ , Fixed $a$ and $z$
§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$
§13.8(iii) Large $a$

§13.8(i) Large $|b|$ , Fixed $a$ and $z$

Notes:: See Slater (1960, §4.3). For (13.8.2) and (13.8.3) use Watson’s lemma for loop integrals (Olver (1997b, §4.5)) and (13.4.10).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.8.i

If $b\to\infty$ in $\Complex$ in such a way that $\left|b+n\right|\geq\delta>0$ for all $n=0,1,2,\dots$ , then

13.8.1 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=\sum _{{s=0}}^{{n-1}}\frac{\left(a\right)_{{s}}}{\left(b\right)_{{s}}s!}z^{{s}}+\mathop{O\/}\nolimits\!\left(|b|^{{-n}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : 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

For fixed $a$ and $z$ in $\Complex$

13.8.2 $\mathop{M\/}\nolimits\!\left(a,b,z\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}\sum _{{s=0}}^{{\infty}}\left(a\right)_{{s}}q_{{s}}(z,a)b^{{-s-a}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\sim$ : asymptotic equality, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.8(i)
Permalink:: http://dlmf.nist.gov/13.8.E2
Encodings:: TeX, pMML, png

as $b\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits b|\leq\pi-\delta$ , where $q_{0}(z,a)=1$ and

13.8.3 $\left(e^{{t}}-1\right)^{{a-1}}\mathop{\exp\/}\nolimits\!\left(t+z(1-e^{{-t}})\right)=\sum _{{s=0}}^{{\infty}}q_{{s}}(z,a)t^{{s+a-1}}.$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.8(i)
Permalink:: http://dlmf.nist.gov/13.8.E3
Encodings:: TeX, pMML, png

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.

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

Notes:: See Temme (1978).
Keywords:: Kummer functions
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/13.8.ii
Addition (effective with 1.0.5):: The reference to López and Pagola (2010) has been added at the end of this subsection.
Reported 2012-07-30

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

13.8.4 $\mathop{M\/}\nolimits\!\left(a,b,z\right)\sim b^{{\frac{1}{2}a}}e^{{\frac{1}{4}\zeta^{2}b}}\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{{a-1}}\mathop{U\/}\nolimits\!\left(a-\tfrac{1}{2},-\zeta\sqrt{b}\right)+\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{{a-1}}-\left(\frac{\zeta}{\lambda-1}\right)^{{a}}\right)\frac{\mathop{U\/}\nolimits\!\left(a-\tfrac{3}{2},-\zeta\sqrt{b}\right)}{\zeta\sqrt{b}}\right)$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\sim$ : asymptotic equality and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.8.E4
Encodings:: TeX, pMML, png

and

13.8.5 $\mathop{U\/}\nolimits\!\left(a,b,z\right)\sim b^{{-\frac{1}{2}a}}e^{{\frac{1}{4}\zeta^{2}b}}\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{{a-1}}\mathop{U\/}\nolimits\!\left(a-\tfrac{1}{2},\zeta\sqrt{b}\right)-\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{{a-1}}-\left(\frac{\zeta}{\lambda-1}\right)^{{a}}\right)\frac{\mathop{U\/}\nolimits\!\left(a-\tfrac{3}{2},\zeta\sqrt{b}\right)}{\zeta\sqrt{b}}\right)$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\sim$ : asymptotic equality and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.8.E5
Encodings:: TeX, pMML, png

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

Special cases are

13.8.6 $\mathop{M\/}\nolimits\!\left(a,b,b\right)=\sqrt{\pi}\left(\frac{b}{2}\right)^{{\frac{1}{2}a}}\left(\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a\right)}+\mathop{O\/}\nolimits\!\left(\frac{1}{b}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function
Permalink:: http://dlmf.nist.gov/13.8.E6
Encodings:: TeX, pMML, png

and

13.8.7 $\mathop{U\/}\nolimits\!\left(a,b,b\right)=\sqrt{\pi}\left(2b\right)^{{-\frac{1}{2}a}}\left(\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a\right)}+\mathop{O\/}\nolimits\!\left(\frac{1}{b}\right)\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function
Permalink:: http://dlmf.nist.gov/13.8.E7
Encodings:: TeX, pMML, png

To obtain approximations for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ that hold as $b\to\infty$ , with $a>\tfrac{1}{2}-b$ 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).

§13.8(iii) Large $a$

Notes:: See Temme (1990b).
Keywords:: Kummer functions
Referenced by:: §2.8(iv)
Permalink:: http://dlmf.nist.gov/13.8.iii

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

When $a\to+\infty$ with $b$ ( $\leq 1$ ) fixed,

13.8.8 $\mathop{U\/}\nolimits\!\left(a,b,x\right)=\frac{2e^{{\frac{1}{2}x}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\left(\sqrt{\frac{2}{\beta}\mathop{\tanh\/}\nolimits\!\left(\frac{w}{2}\right)}\left(\frac{1-e^{{-w}}}{\beta}\right)^{{-b}}\beta^{{1-b}}\mathop{K_{{1-b}}\/}\nolimits\!\left(2\beta a\right)+a^{{-1}}\left(\frac{a^{{-1}}+\beta}{1+\beta}\right)^{{1-b}}e^{{-2\beta a}}\mathop{O\/}\nolimits\!\left(1\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $x$ : real variable
Referenced by:: §13.8(iii)
Permalink:: http://dlmf.nist.gov/13.8.E8
Encodings:: TeX, pMML, png

where $w=\mathop{\mathrm{arccosh}\/}\nolimits\!\left(1+(2a)^{{-1}}x\right)$ , and $\beta=\ifrac{(w+\mathop{\sinh\/}\nolimits w)}{2}$ . (13.8.8) holds uniformly with respect to $x\in[0,\infty)$ . 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 $a\to-\infty$ with $b$ ( $\geq 1$ ) fixed,

13.8.9 $\mathop{M\/}\nolimits\!\left(a,b,x\right)=\mathop{\Gamma\/}\nolimits\!\left(b\right)e^{{\frac{1}{2}x}}\left((\tfrac{1}{2}b-a)x\right)^{{\frac{1}{2}-\frac{1}{2}b}}\*\left(\mathop{J_{{b-1}}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)+\mathop{\mathrm{env}J_{{b-1}}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)\mathop{O\/}\nolimits\!\left(\left|a\right|^{{-\frac{1}{2}}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function 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

and

13.8.10 $\mathop{U\/}\nolimits\!\left(a,b,x\right)=\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b-a+\tfrac{1}{2}\right)e^{{\frac{1}{2}x}}x^{{\frac{1}{2}-\frac{1}{2}b}}\*\left(\mathop{\cos\/}\nolimits\!\left(a\pi\right)\mathop{J_{{b-1}}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)-\mathop{\sin\/}\nolimits\!\left(a\pi\right)\mathop{Y_{{b-1}}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)+\mathop{\mathrm{env}Y_{{b-1}}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)\mathop{O\/}\nolimits\!\left(\left|a\right|^{{-\frac{1}{2}}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : 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

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

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

§13.8 Asymptotic Approximations for Large Parameters

Contents

§13.8(i) Large , Fixed and

§13.8(ii) Large and , Fixed and

§13.8(iii) Large

§13.8(i) Large $|b|$ , Fixed $a$ and $z$

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

§13.8(iii) Large $a$