# §5.11 Asymptotic Expansions

## §5.11(i) Poincaré-Type Expansions

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

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

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

With the same conditions,

 5.11.3 $\Gamma\left(z\right)\sim{\mathrm{e}}^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2% }\sum_{k=0}^{\infty}\frac{g_{k}}{z^{k}},$ ⓘ 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, $k$: 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), §8.12 Permalink: http://dlmf.nist.gov/5.11.E3 Encodings: TeX, pMML, png 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 $\displaystyle g_{0}$ $\displaystyle=1,$ $\displaystyle g_{1}$ $\displaystyle=\tfrac{1}{12},$ $\displaystyle g_{2}$ $\displaystyle=\tfrac{1}{288},$ $\displaystyle g_{3}$ $\displaystyle=-\tfrac{139}{51840},$ $\displaystyle g_{4}$ $\displaystyle=-\tfrac{571}{24\;88320},$ $\displaystyle g_{5}$ $\displaystyle=\tfrac{1\;63879}{2090\;18880},$ $\displaystyle g_{6}$ $\displaystyle=\tfrac{52\;46819}{7\;52467\;96800}.$ ⓘ 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 $g_{k}=\sqrt{2}{\left(\tfrac{1}{2}\right)_{k}}a_{2k},$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $k$: 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 $a_{0}a_{k}+\frac{1}{2}a_{1}a_{k-1}+\frac{1}{3}a_{2}a_{k-2}+\dots+\frac{1}{k+1}% a_{k}a_{0}=\frac{1}{k}a_{k-1},$ $k\geq 1$. ⓘ 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 See also: Annotations for §5.11(i), §5.11 and Ch.5

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

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

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, $k$: nonnegative integer and $z$: 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 $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,~{}1]$. In fact, $h$ 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 $|\Gamma\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{x-(1/2)}e^{-\pi|y|/2},$

uniformly for bounded real values of $x$.

## §5.11(ii) Error Bounds and Exponential Improvement

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

Then

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

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

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

 5.11.12 $\displaystyle\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}$ $\displaystyle\sim z^{a-b},$ 5.11.13 $\displaystyle\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}$ $\displaystyle\sim z^{a-b}\sum_{k=0}^{\infty}\frac{G_{k}(a,b)}{z^{k}},$
 5.11.14 $\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}\sim\left(z+\frac{a+b-1}{% 2}\right)^{a-b}\sum_{k=0}^{\infty}\frac{H_{k}(a,b)}{\left(z+\tfrac{1}{2}(a+b-1% )\right)^{2k}}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\Re$: real part, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: 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 $\displaystyle G_{0}(a,b)$ $\displaystyle=1,$ $\displaystyle G_{1}(a,b)$ $\displaystyle=\tfrac{1}{2}(a-b)(a+b-1),$ $\displaystyle G_{2}(a,b)$ $\displaystyle=\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-b}{2}(3(a+b-1)^{2}-(a-b+1)),$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: 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 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 $\displaystyle H_{0}(a,b)$ $\displaystyle=1,$ $\displaystyle H_{1}(a,b)$ $\displaystyle=-\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-b}{2}(a-b+1),$ $\displaystyle H_{2}(a,b)$ $\displaystyle=\frac{1}{240}\genfrac{(}{)}{0.0pt}{}{a-b}{4}(2(a-b+1)+5(a-b+1)^{% 2}).$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $a$: 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 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 $\displaystyle G_{k}(a,b)$ $\displaystyle=\genfrac{(}{)}{0.0pt}{}{a-b}{k}B^{(a-b+1)}_{k}\left(a\right),$ ⓘ 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, $k$: nonnegative integer, $a$: real or complex variable and $b$: 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 $\displaystyle H_{k}(a,b)$ $\displaystyle=\genfrac{(}{)}{0.0pt}{}{a-b}{2k}B^{(a-b+1)}_{2k}\left(\frac{a-b+% 1}{2}\right).$ ⓘ 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, $k$: nonnegative integer, $a$: real or complex variable and $b$: 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 $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$), then

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

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