§ 5.11. Asymptotic Expansions

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).
Permalink:: http://dlmf.nist.gov/5.11.SS1

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : 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
Encodings:: TeX, pMathML, png

and

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

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $\sim$ : asymptotically equal, $k$ : 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
Encodings:: TeX, pMathML, png

For the Bernoulli numbers $B_{{2k}}$ , see §Ch.24.

With the same conditions,

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

Defines:: $g_{k}$ : coefficients
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.37
Referenced by:: §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21
Permalink:: http://dlmf.nist.gov/5.11.E3
Encodings:: TeX, pMathML, png

where

5.11.4

$g_{0}=1,$

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

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

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

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

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

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

Defines:: $g_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/5.11.E4
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, png, png, png, png, png, png, png

Also,

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

Defines:: $g_{k}$ : coefficients and $a_{k}$ : coefficient
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer's symbol and $k$ : nonnegative integer
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E5
Encodings:: TeX, pMathML, png

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

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

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $z$ : complex variable, $b$ : real or complex variable and $a_{k}$ : coefficient
A&S Ref:: 6.1.39
Referenced by:: §2.10(iii), §2.10(iv), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E7
Encodings:: TeX, pMathML, png

where $a\;(>0)$ and $b\;(\in\Complex)$ are both fixed, and

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E8
Encodings:: TeX, pMathML, png

where $h\;(\in[0,1])$ is fixed, and $B_{{k}}\!\left(h\right)$ is the Bernoulli polynomial defined in §Ch.24.

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $x$ : real variable and $y$ : real variable
Referenced by:: §2.5(i), §5.11(i)
Permalink:: http://dlmf.nist.gov/5.11.E9
Encodings:: TeX, pMathML, png

uniformly for bounded real values of $x$ .

§ 5.11(ii). Error Bounds and Exponential Improvement

Referenced by:: §5.6(i)
Permalink:: http://dlmf.nist.gov/5.11.SS2

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ( $k\ge 0$ ) 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 ${\sec^{{2n}}}\!\left(\tfrac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec^{{2n+1}}}\!\left(\tfrac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms.

For the remainder term in (5.11.3) write

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer, $z$ : complex variable, $g_{k}$ : coefficients and $R_{K}(z)$ : remainder
Permalink:: http://dlmf.nist.gov/5.11.E10
Encodings:: TeX, pMathML, png

Then

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

Defines:: $R_{K}(z)$ : remainder
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.11.E11
Encodings:: TeX, pMathML, png

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

Notes:: See Temme (1996, pp. 67–68), Olver (1997b, p. 119), and Paris and Kaminski (2001, pp. 50–54).
Permalink:: http://dlmf.nist.gov/5.11.SS3

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

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.11.E12
Encodings:: TeX, pMathML, png

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : 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
Encodings:: TeX, pMathML, png

Also, with the added condition $\realpart{(b-a)}>0$ ,

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:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : 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, pMathML, 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:: $a$ : 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, pMathML, pMathML, pMathML, png, png, png

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:: $a$ : 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, pMathML, pMathML, pMathML, png, png, png

In terms of generalized Bernoulli polynomials $B^{{(\ell)}}_{{n}}\!\left(x\right)$ (§Ch.24), we have for $k=0,1,\ldots,$

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

Defines:: $G_{k}(a,b)$ : coefficients
Symbols:: $k$ : nonnegative integer, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.11.E17
Encodings:: TeX, pMathML, png

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

Defines:: $H_{k}(a,b)$ : coefficients
Symbols:: $k$ : nonnegative integer, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.11.E18
Encodings:: TeX, pMathML, png

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\left(a\right)_{{n}}$ : Pochhammer's symbol, $\sim$ : asymptotically equal, $k$ : 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, pMathML, 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