13 Confluent Hypergeometric Functions Kummer Functions 13.7 Asymptotic Expansions for Large Argument 13.9 Zeros

§13.8 Asymptotic Approximations for Large Parameters

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Referenced by:: §13.29(i)
Permalink:: http://dlmf.nist.gov/13.8
See also:: Annotations for Ch.13

§13.8(i) Large , Fixed and
§13.8(ii) Large and , Fixed and
§13.8(iii) Large
§13.8(iv) Large and

§13.8(i) Large , Fixed and

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Keywords:: Kummer functions, asymptotic approximations for large parameters, large
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).
Permalink:: http://dlmf.nist.gov/13.8.i
See also:: Annotations for §13.8 and Ch.13

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : nonnegative integer and : 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 and in $\mathbb{C}$

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, : nonnegative integer and : 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 $b\to\infty$ in $|\operatorname{ph}b|\leq\pi-\delta$ , where $q_{0}(z,a)=1$ and

13.8.3 $\left(e^{t}-1\right)^{a-1}\exp\left(t+z(1-e^{-t})\right)=\sum_{s=0}^{\infty}q_% {s}(z,a)t^{s+a-1}.$

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Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 is large, and and are bounded.

§13.8(ii) Large and , Fixed and

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Keywords:: Kummer functions, asymptotic approximations for large parameters, large , uniform
Notes:: See Temme (1978).
Referenced by:: Erratum (V1.0.11) for References, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/13.8.ii
Addition (effective with 1.0.11):: A reference to Temme (2015, §§10.4 and 22.5) was added at the end of this subsection.
Addition (effective with 1.0.5):: The reference to López and Pagola (2010) has been added at the end of this subsection.
See also:: Annotations for §13.8 and Ch.13

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

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

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Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\sim$ : asymptotic equality, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and : 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 $U\left(a,b,z\right)\sim b^{-\frac{1}{2}a}e^{\frac{1}{4}\zeta^{2}b}\left(% \lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U\left(a-\tfrac{1}{2},\zeta% \sqrt{b}\right)-\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(% \frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U\left(a-\tfrac{3}{2},\zeta% \sqrt{b}\right)}{\zeta\sqrt{b}}\right)$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\sim$ : asymptotic equality, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and : 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 $b\to\infty$ , uniformly in compact $\lambda$ -intervals of $(0,\infty)$ and compact real -intervals. For the parabolic cylinder function see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

Special cases are

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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 $U\left(a,b,b\right)=\sqrt{\pi}\left(2b\right)^{-\frac{1}{2}a}\left(\frac{1}{% \Gamma\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\Gamma\left(\frac% {1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : 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 $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$ that hold as $b\to\infty$ , with $a>\tfrac{1}{2}-b$ and 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 and see López and Pagola (2010).

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

§13.8(iii) Large

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Keywords:: Kummer functions, asymptotic approximations for large parameters, large , uniform
Notes:: See Temme (1990b).
Referenced by:: §2.8(iv), Erratum (V1.0.11) for Subsection 13.8(iii)
Permalink:: http://dlmf.nist.gov/13.8.iii
Addition (effective with 1.0.11):: A paragraph containing new equations (13.8.11)–(13.8.16) and a reference to Temme (2015, Chapters 10 and 27) have been added at the end of this subsection.
See also:: Annotations for §13.8 and Ch.13

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

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\tanh\NVar{z}$ : hyperbolic tangent function, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and : 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 $w=\operatorname{arccosh}\left(1+(2a)^{-1}x\right)$ , and $\beta=\ifrac{(w+\sinh w)}{2}$ . (13.8.8) holds uniformly with respect to $x\in[0,\infty)$ . For the case 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 ( $\geq 1$ ) fixed,

13.8.9 $M\left(a,b,x\right)=\Gamma\left(b\right)e^{\frac{1}{2}x}\left((\tfrac{1}{2}b-a% )x\right)^{\frac{1}{2}-\frac{1}{2}b}\*\left(J_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\operatorname{env}\mskip-2.0muJ_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({% \left|a\right|}^{-\frac{1}{2}}\right)\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\operatorname{env}\mskip-2.0muJ_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm and : 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 $U\left(a,b,x\right)=\Gamma\left(\tfrac{1}{2}b-a+\tfrac{1}{2}\right)e^{\frac{1}% {2}x}x^{\frac{1}{2}-\frac{1}{2}b}\*\left(\cos\left(a\pi\right)J_{b-1}\left(% \sqrt{2x(b-2a)}\right)-\sin\left(a\pi\right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\operatorname{env}\mskip-2.0muY_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({% \left|a\right|}^{-\frac{1}{2}}\right)\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function and : 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 in each case.

For asymptotic approximations to $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ 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).

When $a\to\infty$ in $\left|\operatorname{ph}a\right|\leq\pi-\delta$ and and fixed,

13.8.11 $U\left(a,b,z\right)\sim 2\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}}{\Gamma\left(% a\right)}\*\left(K_{b-1}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(% z)}{a^{s}}+\sqrt{z/a}K_{b}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s% }(z)}{a^{s}}\right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, : nonnegative integer and : 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 ${\mathbf{M}}\left(a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}% \Gamma\left(1+a-b\right)}{\Gamma\left(a\right)}\*\left(I_{b-1}\left(2\sqrt{az}% \right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}-\sqrt{z/a}I_{b}\left(2\sqrt{% az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{a^{s}}\right),$

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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, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : nonnegative integer and : 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 ${\mathbf{M}}\left(-a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}% \Gamma\left(1+a\right)}{\Gamma\left(a+b\right)}\*\left(J_{b-1}\left(2\sqrt{az}% \right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-\sqrt{z/a}J_{b}\left(2% \sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{(-a)^{s}}\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $U\left(-a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}e^{z/2}\Gamma\left(1+a\right% )\*\left(C_{b-1}(a,2\sqrt{az})\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-% \sqrt{z/a}C_{b}(a,2\sqrt{az})\sum_{s=0}^{\infty}\frac{q_{s}(z)}{(-a)^{s}}% \right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $(\NVar{a},\NVar{b})$ : open interval, : nonnegative integer and : 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 $C_{\nu}\left(a,\zeta\right)=\cos\left(\pi a\right)J_{\nu}\left(\zeta\right)+% \sin\left(\pi a\right)Y_{\nu}\left(\zeta\right)$ and

13.8.15

$p_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}{s}{\left(1-b+s\right)_{k-s}}% z^{s}c_{k+s}(z),$

$q_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}{s}{\left(2-b+s\right)_{k-s}}% z^{s}c_{k+s+1}(z)$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : nonnegative integer and : 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 $(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{bB_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2% }}{(s+2)!}\right)c_{k-s}(z)=0,$ $k=0,1,2,\dots$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $(\NVar{a},\NVar{b})$ : open interval, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : nonnegative integer and : 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 for $c_{k}(z)$ is given. It satisfies,
$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{e^{t}-1}\right% )-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-1\right)^{2}}\right)\right)f.$
For (13.8.16) combine this differential equation with $f(z,t)=\sum_{k=0}^{\infty}c_{k}(z)t^{k}$ 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 can also be unbounded see Temme (2015, Chapters 10 and 27)

§13.8(iv) Large and

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Keywords:: Kummer functions, asymptotic approximations for large parameters, large and
Referenced by:: Erratum (V1.1.7) for Subsection 13.8(iv), Erratum (V1.1.7) for Subsection 13.8(iv)
Permalink:: http://dlmf.nist.gov/13.8.iv
Addition (effective with 1.1.7):: This subsection was added.
See also:: Annotations for §13.8 and Ch.13

When $a,b\to+\infty$ with $\left|z\right|$ and $\nu=\frac{a}{b}$ bounded

13.8.17 $M\left(a,b,z\right)={\mathrm{e}}^{\nu z}\frac{\Gamma^{*}(b)}{\Gamma^{*}(a)}% \left(1+\frac{(1-\nu)(1+6\nu^{2}z^{2})}{12a}+O\left(\frac{1}{\min(a^{2},b^{2})% }\right)\right),$

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and : complex variable
Permalink:: http://dlmf.nist.gov/13.8.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §13.8(iv), §13.8 and Ch.13

13.8.18 $U\left(a,b+1,z\right)=z^{-b}{\mathrm{e}}^{(1-\nu)z}\frac{\Gamma\left(b\right)}% {\Gamma\left(a\right)}\left(1+\frac{\nu z(1-\nu)(2-\nu z)}{2a}+O\left(\frac{1}% {\min(a^{2},b^{2})}\right)\right),$ $\Re z>0$ ,

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\Re$ : real part and : complex variable
Permalink:: http://dlmf.nist.gov/13.8.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §13.8(iv), §13.8 and Ch.13

where $\Gamma^{*}\left(a\right)$ is the scaled gamma function defined in (5.11.3). These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. For generalizations in which is also allowed to be large see Temme and Veling (2022).

13.7 Asymptotic Expansions for Large Argument 13.9 Zeros

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§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(iv) Large and