DLMF: GA.11 Asymptotic Expansions

§GA.11. Asymptotic Expansions

Sources:

Olver(1997)(Chapter 3, §8, Chapter 4, §5, and Chapter 8, §4), Temme(1996)(§3.6.2), Paris and Kaminski(2001)(§2.2.5). (GA.11.7) and (GA.11.9) are derived from (GA.11.3).

§GA.11(i)Poincaré-Type Expansions
§GA.11(ii)Error Bounds and Exponential Improvement
§GA.11(iii)Ratios

§GA.11(i). Poincaré-Type Expansions

Notes:

Olver(1997)(pp. 87–88 and 293–295)

Keywords:

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

GA.11.1 $\mathrm{ln}\Gamma\!\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\mathrm{ln}z-z+\tfrac{1}{2}\mathrm{ln}\!\left(2\pi\right)+\sum _{{k=1}}^{\infty}\frac{B_{{2k}}}{2k(2k-1)z^{{2k-1}}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.40
Referenced by:: §GA.11(i), §GA.11(ii), §GA.11(ii)
Encodings:: pMathML, png, TeX

GA.11.2 $\psi\!\left(z\right)\sim\mathrm{ln}z-\frac{1}{2z}-\sum _{{k=1}}^{\infty}\frac{B_{{2k}}}{2kz^{{2k}}}.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.3.18
Referenced by:: §GA.11(ii), §GA.4(iii), §GA.4
Encodings:: pMathML, png, TeX

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

GA.11.3 $\Gamma\!\left(z\right)\sim e^{{-z}}z^{z}\left(\frac{2\pi}{z}\right)^{{1/2}}\left(\sum _{{k=0}}^{\infty}\frac{g_{k}}{z^{k}}\right).$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer, $z$ : complex variable and $g_{k}$ : coefficients
A&S Ref:: 6.1.37
Referenced by:: §GA.11(i), §GA.11(ii), §GA.11(ii), §GA.11, §GA.21
Encodings:: pMathML, png, TeX

GA.11.4 $g_{0}=1,$ $g_{1}=\tfrac{1}{12},$ $g_{2}=\tfrac{1}{288},$ $g_{3}=-\tfrac{139}{51840},$ $g_{4}=-\tfrac{571}{24\; 88320},$ $g_{5}=\tfrac{1\; 63879}{2090\; 18880},$ $g_{6}=\tfrac{52\; 46819}{7\; 52467\; 96800}.$

Notations:: $g_{k}$ : coefficients
Encodings:: pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, png, png, png, png, png, png, png, TeX, TeX, TeX, TeX, TeX, TeX, TeX

GA.11.5 $g_{k}=\sqrt{2}\left(\tfrac{1}{2}\right)_{{k}}a_{{2k}},$

Notations:: $\left(a\right)_{{n}}$ : Pochhammer's symbol, $k$ : nonnegative integer, $g_{k}$ : coefficients and $a_{k}$ : coefficient
Referenced by:: §GA.11(i)
Encodings:: pMathML, png, TeX

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

GA.11.6 $a_{0}a_{k}+\tfrac{1}{2}a_{1}a_{{k-1}}+\tfrac{1}{3}a_{2}a_{{k-2}}+\dots+\tfrac{1}{k+1}a_{k}a_{0}=\tfrac{1}{k}a_{{k-1}},$ $k\ge 1$ .

Notations:: $k$ : nonnegative integer and $a_{k}$ : coefficient
Referenced by:: §GA.11(i)
Encodings:: pMathML, png, TeX

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).

With the same conditions

GA.11.7 $\Gamma\!\left(az+b\right)\sim\sqrt{2\pi}e^{{-az}}(az)^{{az+b-(1/2)}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable, $b$ : real or complex variable and $a_{k}$ : coefficient
A&S Ref:: 6.1.39
Referenced by:: §GA.11(i), §GA.11
Encodings:: pMathML, png, TeX

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

GA.11.8 $\mathrm{ln}\Gamma\!\left(z+h\right)\sim\left(z+h-\tfrac{1}{2}\right)\mathrm{ln}z-z+\tfrac{1}{2}\mathrm{ln}\!\left(2\pi\right)+\sum _{{k=2}}^{\infty}\frac{(-1)^{k}B_{{k}}\!\left(h\right)}{k(k-1)z^{{k-1}}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §GA.11(i)
Encodings:: pMathML, png, TeX

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

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

GA.11.9 $|\Gamma\!\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{{x-(1/2)}}e^{{-\pi|y|/2}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable and $y$ : real variable
Referenced by:: §GA.11(i), §GA.11
Encodings:: pMathML, png, TeX

uniformly for bounded real values of $x$ .

§GA.11(ii). Error Bounds and Exponential Improvement

Keywords:: gamma function
Referenced by:: §GA.6(i), §GA.6

If the sums in the expansions (GA.11.1) and (GA.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 ${\mathrm{sec}^{{2n}}}\!\left(\tfrac{1}{2}\mathrm{ph}z\right)$ for (GA.11.1), and ${\mathrm{sec}^{{2n+1}}}\!\left(\tfrac{1}{2}\mathrm{ph}z\right)$ for (GA.11.2), times the first neglected terms.

For the remainder term in (GA.11.3) write

GA.11.10 $\Gamma\!\left(z\right)=e^{{-z}}z^{z}\left(\frac{2\pi}{z}\right)^{{1/2}}\left(\sum _{{k=0}}^{{K-1}}\frac{g_{k}}{z^{k}}+R_{K}(z)\right),$ $K=1,2,3,\dots$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
Encodings:: pMathML, png, TeX

Then

GA.11.11 $\left|R_{K}(z)\right|\leq\frac{(1+\zeta\!\left(K\right))\Gamma\!\left(K\right)}{2(2\pi)^{{K+1}}\left|z\right|^{K}}\*\left(1+\min(\mathrm{sec}\!\left(\mathrm{ph}z\right),2K^{{\frac{1}{2}}})\right),$ $\left|\mathrm{ph}z\right|\leq\frac{1}{2}\pi$ ,

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Encodings:: pMathML, png, TeX

where $\zeta\!\left(K\right)$ is as in Chapter ZE. 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(1997)(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 (GA.11.1) and (GA.11.3) in series of incomplete gamma functions with exponential improvement (§) in the asymptotic expansions, see Berry(1991), Boyd(1994), and Paris and Kaminski(2001)(§6.4).

§GA.11(iii). Ratios

Notes:

Temme(1996)(pp. 67–68)

Olver(1997)(p. 119)

Paris and Kaminski(2001)(pp. 50–54)

Keywords:

gamma function

If $a\;(\in\Complex)$ and $b\;(\in\Complex)$ are fixed as $z\to\infty$ in $|\mathrm{ph}z|\le\pi-\delta\;(<\pi)$ , then

GA.11.12 $\frac{\Gamma\!\left(z+a\right)}{\Gamma\!\left(z+b\right)}\sim z^{{a-b}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Encodings:: pMathML, png, TeX

GA.11.13 $\frac{\Gamma\!\left(z+a\right)}{\Gamma\!\left(z+b\right)}\sim z^{{a-b}}\sum _{{k=0}}^{\infty}\frac{G_{k}(a,b)}{z^{k}}.$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : 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
Encodings:: pMathML, png, TeX

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

GA.11.14 $\frac{\Gamma\!\left(z+a\right)}{\Gamma\!\left(z+b\right)}\sim\left(z+\frac{a+b-1}{2}\right)^{{a-b}}\sum _{{k=0}}^{\infty}\frac{H_{k}(a,b)}{\left(z+\tfrac{1}{2}(a+b-1)\right)^{{2k}}}.$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex variable, $b$ : real or complex variable and $H_{k}(a,b)$ : coefficients
Encodings:: pMathML, png, TeX

Here

GA.11.15 $G_{0}(a,b)=1,$ $G_{1}(a,b)=\tfrac{1}{2}(a-b)(a+b-1),$ $G_{2}(a,b)=\frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1)),$

Notations:: $a$ : real or complex variable, $b$ : real or complex variable and $G_{k}(a,b)$ : coefficients
Encodings:: pMathML, pMathML, pMathML, png, png, png, TeX, TeX, TeX

GA.11.16 $H_{0}(a,b)=1,$ $H_{1}(a,b)=-\frac{1}{12}\binom{a-b}{2}(a-b+1),$ $H_{2}(a,b)=\frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2}).$

Notations:: $a$ : real or complex variable, $b$ : real or complex variable and $H_{k}(a,b)$ : coefficients
Encodings:: pMathML, pMathML, pMathML, png, png, png, TeX, TeX, TeX

In terms of generalized Bernoulli polynomials (§) we have for $k=0,1,\ldots,$

GA.11.17 $G_{k}(a,b)=\binom{a-b}{k}B^{{(a-b+1)}}_{{k}}\!\left(a\right),$

Notations:: $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $G_{k}(a,b)$ : coefficients
Encodings:: pMathML, pMathML, png, png, TeX, TeX

GA.11.18 $H_{k}(a,b)=\binom{a-b}{2k}B^{{(a-b+1)}}_{{2k}}\!\left(\frac{a-b+1}{2}\right).$

Notations:: $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $H_{k}(a,b)$ : coefficients
Encodings:: pMathML, pMathML, png, png, TeX, TeX

GA.11.19 $\frac{\Gamma\!\left(z+a\right)\Gamma\!\left(z+b\right)}{\Gamma\!\left(z+c\right)}\sim\sum _{{k=0}}^{\infty}(-1)^{k}\frac{\left(c-a\right)_{{k}}\left(c-b\right)_{{k}}}{k!}\Gamma\!\left(a+b-c+z-k\right).$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $\left(a\right)_{{n}}$ : Pochhammer's symbol, $k$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.11(iii)
Encodings:: pMathML, png, TeX

For the error term in (GA.11.19) in the case $z=x\;(>0)$ and $c=1$ , see Olver(1995).

§GA.11. Asymptotic Expansions

Contents

§GA.11(i). Poincaré-Type Expansions

§GA.11(ii). Error Bounds and Exponential Improvement

§GA.11(iii). Ratios