5 Gamma Function Properties 5.10 Continued Fractions 5.12 Beta Function

§5.11 Asymptotic Expansions

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Keywords:: asymptotic expansions, gamma function
Permalink:: http://dlmf.nist.gov/5.11
See also:: Annotations for Ch.5

§5.11(i) Poincaré-Type Expansions
§5.11(ii) Error Bounds and Exponential Improvement
§5.11(iii) Ratios

§5.11(i) Poincaré-Type Expansions

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Keywords:: asymptotic expansion, psi function
Notes:: See Olver (1997b, pp. 87–88 and 293–295) for (5.11.1)–(5.11.3) and (5.11.5)–(5.11.6). (5.11.7) and (5.11.9) are derived from (5.11.3). For (5.11.8) see Rowe (1931).
Referenced by:: §33.10(ii), §33.10(iii), §4.13, §5.11(i), Erratum (V1.0.10) for References, Erratum (V1.0.11) for References, Erratum (V1.0.11) for Clarifications
Permalink:: http://dlmf.nist.gov/5.11.i
Addition (effective with 1.1.7):: In (5.11.3), the definition of the scaled gamma function $\Gamma^{*}\left(z\right)$ was inserted after the first equals sign. Some descriptive text about the main property of $\Gamma^{*}\left(z\right)$ was inserted at the bottom of the section.
Clarification (effective with 1.1.7):: On the first line of §5.11(i), and just above both (5.11.12), (5.11.19), $|\operatorname{ph}z|\leq\pi-\delta$ ( $<\pi$ ) was changed to $|\operatorname{ph}z|\leq\pi-\delta$ .
Additions and Changes (effective with 1.0.11):: A reference to Rowe (1931) was added in the Notes above. References to Nemes (2013a, 2015a) were added after (5.11.6). The reference to Nemes (2013c) that was added in Release 1.0.10 was deleted and moved in revised form to appear after (5.11.8).
Addition (effective with 1.0.10):: The reference to Nemes (2013c) was added at the end of (5.11.2).
See also:: Annotations for §5.11 and Ch.5

As $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ ,

5.11.1 $\operatorname{Ln}\Gamma\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2% k-1}}$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : complex variable
A&S Ref:: 6.1.40
Referenced by:: §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
Permalink:: http://dlmf.nist.gov/5.11.E1
Encodings:: TeX, pMML, png
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.11(i), §5.11 and Ch.5

and

5.11.2 $\psi\left(z\right)\sim\ln z-\frac{1}{2z}-\sum_{k=1}^{\infty}\frac{B_{2k}}{2kz^% {2k}}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : Poincaré asymptotic expansion, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : complex variable
A&S Ref:: 6.3.18
Referenced by:: §5.11(i), §5.11(ii), §5.4(iii)
Permalink:: http://dlmf.nist.gov/5.11.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(i), §5.11 and Ch.5

For the Bernoulli numbers $B_{2k}$ , see §24.2(i).

With the same conditions,

5.11.3 $\Gamma\left(z\right)={\mathrm{e}}^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}% \Gamma^{*}\left(z\right)\sim{\mathrm{e}}^{-z}z^{z}\left(\frac{2\pi}{z}\right)^% {1/2}\sum_{k=0}^{\infty}\frac{g_{k}}{z^{k}},$

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Defines:: $\Gamma^{*}\left(\NVar{z}\right)$ : scaled gamma function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : nonnegative integer, : complex variable and $g_{k}$ : coefficients
A&S Ref:: 6.1.37
Referenced by:: §10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement
Permalink:: http://dlmf.nist.gov/5.11.E3
Encodings:: TeX, pMML, png
Addition (effective with 1.1.7):: The definition of the scaled gamma function $\Gamma^{*}\left(z\right)$ was inserted after the first equals sign.
Changes (effective with 1.0.11):: An unnecessary bracket around the sum was removed.
See also:: Annotations for §5.11(i), §5.11 and Ch.5

where

5.11.4

$g_{0}=1,$

$g_{1}=\tfrac{1}{12},$

$g_{2}=\tfrac{1}{288},$

$g_{3}=-\tfrac{139}{51840},$

$g_{4}=-\tfrac{571}{24\;88320},$

$g_{5}=\tfrac{1\;63879}{2090\;18880},$

$g_{6}=\tfrac{52\;46819}{7\;52467\;96800}.$

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Symbols:: $g_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E4
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png
See also:: Annotations for §5.11(i), §5.11 and Ch.5

Also,

5.11.5 $g_{k}=\sqrt{2}{\left(\tfrac{1}{2}\right)_{k}}a_{2k},$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : nonnegative integer, $g_{k}$ : coefficients and $a_{k}$ : coefficient
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(i), §5.11 and Ch.5

where $a_{0}=\tfrac{1}{2}\sqrt{2}$ and

5.11.6 $a_{0}a_{k}+\frac{1}{2}a_{1}a_{k-1}+\frac{1}{3}a_{2}a_{k-2}+\dots+\frac{1}{k+1}% a_{k}a_{0}=\frac{1}{k}a_{k-1},$ $k\geq 1$ .

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Symbols:: : nonnegative integer and $a_{k}$ : coefficient
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(i), §5.11 and Ch.5

The scaled gamma function $\Gamma^{*}\left(z\right)$ is defined in (5.11.3) and its main property is $\Gamma^{*}\left(z\right)\sim 1$ as $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ . Wrench (1968) gives exact values of $g_{k}$ up to $g_{20}$ . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of $g_{k}$ for $k=21,22,\dots,30$ . For explicit formulas for $g_{k}$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_{k}$ as $k\to\infty$ see Boyd (1994) and Nemes (2015a).

Terminology

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Keywords:: Stirling’s formula, Stirling’s series
See also:: Annotations for §5.11(i), §5.11 and Ch.5

The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)).

Next, and again with the same conditions,

5.11.7 $\Gamma\left(az+b\right)\sim\sqrt{2\pi}e^{-az}(az)^{az+b-(1/2)},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : complex variable, : real or complex variable and $a_{k}$ : coefficient
A&S Ref:: 6.1.39
Referenced by:: §11.6(ii), §13.21(i), §2.10(iii), §2.10(iv), §26.3(v), §33.5(iv), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

where $a\;(>0)$ and $b\;(\in\mathbb{C})$ are both fixed, and

5.11.8 $\operatorname{Ln}\Gamma\left(z+h\right)\sim\left(z+h-\tfrac{1}{2}\right)\ln z-% z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=2}^{\infty}\frac{(-1)^{k}B_{k}\left% (h\right)}{k(k-1)z^{k-1}},$

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Referenced by:: §12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)
Permalink:: http://dlmf.nist.gov/5.11.E8
Encodings:: TeX, pMML, png
Generalization (effective with 1.0.11):: Originally in (5.11.8) was unnecessarily restricted to lie in the interval . In fact, may lie anywhere in the complex plane.
Suggested 2015-02-28 by Nico Temme
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z+h\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

where $h\;(\in\mathbb{C})$ is fixed, and $B_{k}\left(h\right)$ is the Bernoulli polynomial defined in §24.2(i). For similar results including a convergent factorial series see, Nemes (2013c).

Lastly, as $y\to\pm\infty$ ,

5.11.9 $|\Gamma\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{x-(1/2)}e^{-\pi|y|/2},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable and : real variable
Referenced by:: §16.5, §2.5(i), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

uniformly for bounded real values of .

§5.11(ii) Error Bounds and Exponential Improvement

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Keywords:: asymptotic expansions, error bounds, exponentially-improved, gamma function
Referenced by:: §5.6(i), Erratum (V1.0.10) for References, Erratum (V1.0.11) for References, Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/5.11.ii
Changes (effective with 1.0.11):: The reference to Nemes (2013b) that was added in Release 1.0.10 was deleted and moved in revised form to appear at the end of the first paragraph of this subsection.
Addition (effective with 1.0.10):: A reference to Nemes (2013b) has been added at the end of this section.
Addition (effective with 1.0.7):: A sentence dealing with the special case in (5.11.11) has been added.
Suggested 2013-08-30 by Gergő Nemes
See also:: Annotations for §5.11 and Ch.5

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at ( $k\geq 0$ ) and is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If is complex, then the remainder terms are bounded in magnitude by ${\sec}^{2n}\left(\tfrac{1}{2}\operatorname{ph}z\right)$ for (5.11.1), and ${\sec}^{2n+1}\left(\tfrac{1}{2}\operatorname{ph}z\right)$ for (5.11.2), times the first neglected terms. For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b).

For the remainder term in (5.11.3) write

5.11.10 $\Gamma\left(z\right)=e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k% =0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right),$ $K=1,2,3,\dots$ .

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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, : nonnegative integer, : complex variable, $g_{k}$ : coefficients and $R_{K}(z)$ : remainder
Permalink:: http://dlmf.nist.gov/5.11.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(ii), §5.11 and Ch.5

Then

5.11.11 $\left|R_{K}(z)\right|\leq\frac{(1+\zeta\left(K\right))\Gamma\left(K\right)}{2(% 2\pi)^{K+1}{\left|z\right|}^{K}}\*\left(1+\min(\sec\left(\operatorname{ph}z% \right),2K^{\frac{1}{2}})\right),$ $\left|\operatorname{ph}z\right|\leq\frac{1}{2}\pi$ ,

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Defines:: $R_{K}(z)$ : remainder (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\sec\NVar{z}$ : secant function and : complex variable
Referenced by:: §5.11(ii)
Permalink:: http://dlmf.nist.gov/5.11.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(ii), §5.11 and Ch.5

where $\zeta\left(K\right)$ is as in Chapter 25. In the case the factor $1+\zeta\left(K\right)$ is replaced with 4. For this result and a similar bound for the sector $\frac{1}{2}\pi\leq\operatorname{ph}z\leq\pi$ see Boyd (1994).

For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990).

For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4).

§5.11(iii) Ratios

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Keywords:: asymptotic expansions, for ratios, gamma function
Notes:: See Temme (1996b, pp. 67–68), Olver (1997b, p. 119), Paris and Kaminski (2001, pp. 50–54), and Fields (1966).
Referenced by:: Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/5.11.iii
Addition (effective with 1.0.7):: References to Frenzen (1987a, 1992) and Burić and Elezović (2011) have been added after (5.11.18)
Suggested 2013-08-01 by Gergő Nemes
See also:: Annotations for §5.11 and Ch.5

In this subsection , , and are real or complex constants.

If $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ , then

5.11.12 $\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}\sim z^{a-b},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, : complex variable, : real or complex variable and : real or complex variable
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

5.11.13 $\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}\sim z^{a-b}\sum_{k=0}^{% \infty}\frac{G_{k}(a,b)}{z^{k}},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, : nonnegative integer, : complex variable, : real or complex variable, : real or complex variable and $G_{k}(a,b)$ : coefficients
A&S Ref:: 6.1.47
Permalink:: http://dlmf.nist.gov/5.11.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

5.11.14 $\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}\sim\left(z+\frac{a+b-1}{% 2}\right)^{a-b}\sum_{k=0}^{\infty}\frac{H_{k}(a,b)}{\left(z+\tfrac{1}{2}(a+b-1% )\right)^{2k}}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\Re$ : real part, : nonnegative integer, : complex variable, : real or complex variable, : real or complex variable and $H_{k}(a,b)$ : coefficients
Referenced by:: §5.11(iii), Erratum (V1.0.21) for Equation (5.11.14)
Permalink:: http://dlmf.nist.gov/5.11.E14
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.21):: The previous constraint $\Re\left(b-a\right)>0$ was removed, see Fields (1966, (3)).
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

Here

5.11.15

$G_{0}(a,b)=1,$

$G_{1}(a,b)=\tfrac{1}{2}(a-b)(a+b-1),$

$G_{2}(a,b)=\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-b}{2}(3(a+b-1)^{2}-(a-b+1)),$

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : real or complex variable, : real or complex variable and $G_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E15
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

5.11.16

$H_{0}(a,b)=1,$

$H_{1}(a,b)=-\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-b}{2}(a-b+1),$

$H_{2}(a,b)=\frac{1}{240}\genfrac{(}{)}{0.0pt}{}{a-b}{4}(2(a-b+1)+5(a-b+1)^{2}).$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : real or complex variable, : real or complex variable and $H_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E16
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

In terms of generalized Bernoulli polynomials $B^{(\ell)}_{n}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\ldots$ ,

5.11.17 $G_{k}(a,b)=\genfrac{(}{)}{0.0pt}{}{a-b}{k}B^{(a-b+1)}_{k}\left(a\right),$

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Defines:: $G_{k}(a,b)$ : coefficients (locally)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, : nonnegative integer, : real or complex variable and : real or complex variable
Permalink:: http://dlmf.nist.gov/5.11.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

5.11.18 $H_{k}(a,b)=\genfrac{(}{)}{0.0pt}{}{a-b}{2k}B^{(a-b+1)}_{2k}\left(\frac{a-b+1}{% 2}\right).$

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Defines:: $H_{k}(a,b)$ : coefficients (locally)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, : nonnegative integer, : real or complex variable and : real or complex variable
Referenced by:: §5.11(iii)
Permalink:: http://dlmf.nist.gov/5.11.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). See also Burić and Elezović (2011).

Lastly, and again if $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ , then

5.11.19 $\frac{\Gamma\left(z+a\right)\Gamma\left(z+b\right)}{\Gamma\left(z+c\right)}% \sim\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left(c-a\right)_{k}}{\left(c-b\right)_{% k}}}{k!}\Gamma\left(a+b-c+z-k\right).$

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, : factorial (as in ), : nonnegative integer, : complex variable, : real or complex variable and : real or complex variable
Referenced by:: §5.11(i), §5.11(iii)
Permalink:: http://dlmf.nist.gov/5.11.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §5.11(iii), §5.11 and Ch.5

For the error term in (5.11.19) in the case $z=x\;(>0)$ and , see Olver (1995).

5.10 Continued Fractions 5.12 Beta Function

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§5.11 Asymptotic Expansions

Contents

§5.11(i) Poincaré-Type Expansions

Terminology

§5.11(ii) Error Bounds and Exponential Improvement

§5.11(iii) Ratios