§5.11 Asymptotic Expansions

Keywords:: gamma function
Permalink:: http://dlmf.nist.gov/5.11

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

§5.11(i) Poincaré-Type Expansions

Notes:: See Olver (1997b, pp. 87–88 and 293–295) for (5.11.1)–(5.11.3), (5.11.5)–(5.11.6), and (5.11.8). (5.11.7) and (5.11.9) are derived from (5.11.3).
Keywords:: psi function
Referenced by:: §33.10(ii), §33.10(iii), §4.13
Permalink:: http://dlmf.nist.gov/5.11.i

As $z\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta\;(<\pi)$ ,

5.11.1 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\mathop{\ln\/}\nolimits z-z+\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(2\pi\right)+\sum _{{k=1}}^{\infty}\frac{\mathop{B_{{2k}}\/}\nolimits}{2k(2k-1)z^{{2k-1}}}$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.40
Referenced by:: ¶ ‣ §5.11(i), §5.11(i), §5.11(ii), §5.11(ii)
Permalink:: http://dlmf.nist.gov/5.11.E1
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and

5.11.2 $\mathop{\psi\/}\nolimits\!\left(z\right)\sim\mathop{\ln\/}\nolimits z-\frac{1}{2z}-\sum _{{k=1}}^{\infty}\frac{\mathop{B_{{2k}}\/}\nolimits}{2kz^{{2k}}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.3.18
Referenced by:: §5.11(ii), §5.4(iii)
Permalink:: http://dlmf.nist.gov/5.11.E2
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For the Bernoulli numbers $\mathop{B_{{2k}}\/}\nolimits$ , see §24.2(i).

With the same conditions,

5.11.3 $\mathop{\Gamma\/}\nolimits\!\left(z\right)\sim e^{{-z}}z^{z}\left(\frac{2\pi}{z}\right)^{{1/2}}\left(\sum _{{k=0}}^{\infty}\frac{g_{k}}{z^{k}}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable and $g_{k}$ : coefficients
A&S Ref:: 6.1.37
Referenced by:: §10.19(i), ¶ ‣ §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §8.11(ii), §8.11(v), §8.12
Permalink:: http://dlmf.nist.gov/5.11.E3
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where

5.11.4

$g_{0}=1,$

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

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

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

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

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

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

Symbols:: $g_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E4
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Also,

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $k$ : nonnegative integer, $g_{k}$ : coefficients and $a_{k}$ : coefficient
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E5
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where $a_{0}=\tfrac{1}{2}\sqrt{2}$ and

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

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

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 an asymptotic expansion of $g_{k}$ as $k\to\infty$ see Boyd (1994).

¶ Terminology

Keywords:: Stirling’s formula, Stirling’s series

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $z$ : complex variable, $b$ : 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
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where $a\;(>0)$ and $b\;(\in\Complex)$ are both fixed, and

5.11.8 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z+h\right)\sim\left(z+h-\tfrac{1}{2}\right)\mathop{\ln\/}\nolimits z-z+\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(2\pi\right)+\sum _{{k=2}}^{\infty}\frac{(-1)^{k}\mathop{B_{{k}}\/}\nolimits\!\left(h\right)}{k(k-1)z^{{k-1}}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §12.10(ii), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E8
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where $h\;(\in[0,1])$ is fixed, and $\mathop{B_{{k}}\/}\nolimits\!\left(h\right)$ is the Bernoulli polynomial defined in §24.2(i).

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $x$ : real variable and $y$ : real variable
Referenced by:: §16.5, ¶ ‣ §2.5(i), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E9
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uniformly for bounded real values of $x$ .

§5.11(ii) Error Bounds and Exponential Improvement

Keywords:: gamma function
Referenced by:: §5.6(i)
Permalink:: http://dlmf.nist.gov/5.11.ii

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ( $k\geq 0$ ) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If $z$ is complex, then the remainder terms are bounded in magnitude by ${\mathop{\sec\/}\nolimits^{{2n}}}\!\left(\tfrac{1}{2}\mathop{\mathrm{ph}\/}\nolimits z\right)$ for (5.11.1), and ${\mathop{\sec\/}\nolimits^{{2n+1}}}\!\left(\tfrac{1}{2}\mathop{\mathrm{ph}\/}\nolimits z\right)$ for (5.11.2), times the first neglected terms.

For the remainder term in (5.11.3) write

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $k$ : nonnegative integer, $z$ : complex variable, $g_{k}$ : coefficients and $R_{K}(z)$ : remainder
Permalink:: http://dlmf.nist.gov/5.11.E10
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Then

5.11.11 $\left|R_{K}(z)\right|\leq\frac{(1+\mathop{\zeta\/}\nolimits\!\left(K\right))\mathop{\Gamma\/}\nolimits\!\left(K\right)}{2(2\pi)^{{K+1}}\left|z\right|^{K}}\*\left(1+\min(\mathop{\sec\/}\nolimits\!\left(\mathop{\mathrm{ph}\/}\nolimits z\right),2K^{{\frac{1}{2}}})\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\frac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sec\/}\nolimits z$ : secant function, $z$ : complex variable and $R_{K}(z)$ : remainder
Permalink:: http://dlmf.nist.gov/5.11.E11
Encodings:: TeX, pMML, png

where $\mathop{\zeta\/}\nolimits\!\left(K\right)$ is as in Chapter 25. For this result and a similar bound for the sector $\frac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ 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

Notes:: See Temme (1996b, pp. 67–68), Olver (1997b, p. 119), and Paris and Kaminski (2001, pp. 50–54).
Keywords:: gamma function
Permalink:: http://dlmf.nist.gov/5.11.iii

In this subsection $a$ , $b$ , and $c$ are real or complex constants.

If $z\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ( $<\pi$ ), then

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\sim$ : asymptotic equality, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.11.E12
Encodings:: TeX, pMML, png

5.11.13 $\frac{\mathop{\Gamma\/}\nolimits\!\left(z+a\right)}{\mathop{\Gamma\/}\nolimits\!\left(z+b\right)}\sim z^{{a-b}}\sum _{{k=0}}^{\infty}\frac{G_{k}(a,b)}{z^{k}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex variable, $b$ : real or complex variable and $G_{k}(a,b)$ : coefficients
A&S Ref:: 6.1.47
Permalink:: http://dlmf.nist.gov/5.11.E13
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Also, with the added condition $\realpart{(b-a)}>0$ ,

5.11.14 $\frac{\mathop{\Gamma\/}\nolimits\!\left(z+a\right)}{\mathop{\Gamma\/}\nolimits\!\left(z+b\right)}\sim\left(z+\frac{a+b-1}{2}\right)^{{a-b}}\sum _{{k=0}}^{\infty}\frac{H_{k}(a,b)}{\left(z+\tfrac{1}{2}(a+b-1)\right)^{{2k}}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex variable, $b$ : real or complex variable and $H_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E14
Encodings:: TeX, pMML, png

Here

5.11.15

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

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

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

Symbols:: $\binom{m}{n}$ : binomial coefficient, $a$ : real or complex variable, $b$ : real or complex variable and $G_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E15
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5.11.16

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

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

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

Symbols:: $\binom{m}{n}$ : binomial coefficient, $a$ : real or complex variable, $b$ : 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

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

5.11.17 $G_{k}(a,b)=\binom{a-b}{k}\mathop{B^{{(a-b+1)}}_{{k}}\/}\nolimits\!\left(a\right),$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $G_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E17
Encodings:: TeX, pMML, png

5.11.18 $H_{k}(a,b)=\binom{a-b}{2k}\mathop{B^{{(a-b+1)}}_{{2k}}\/}\nolimits\!\left(\frac{a-b+1}{2}\right).$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $H_{k}(a,b)$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E18
Encodings:: TeX, pMML, png

Lastly, and again if $z\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ( $<\pi$ ), then

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.11(iii)
Permalink:: http://dlmf.nist.gov/5.11.E19
Encodings:: TeX, pMML, png

For the error term in (5.11.19) in the case $z=x\;(>0)$ and $c=1$ , see Olver (1995).

§5.11 Asymptotic Expansions

Contents

§5.11(i) Poincaré-Type Expansions

¶ Terminology

§5.11(ii) Error Bounds and Exponential Improvement

§5.11(iii) Ratios