# §5.11(i) Poincaré-Type Expansions

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

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

and

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

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

With the same conditions,

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

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

Also,

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

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

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

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

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

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

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

Then

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

where $\mathop{\zeta\/}\nolimits\!\left(K\right)$ is as in Chapter 25. In the case $K=1$ the factor $1+\mathop{\zeta\/}\nolimits\!\left(K\right)$ is replaced with 4. 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

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

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

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

Here

 5.11.15 $\displaystyle G_{0}(a,b)$ $\displaystyle=1,$ $\displaystyle G_{1}(a,b)$ $\displaystyle=\tfrac{1}{2}(a-b)(a+b-1),$ $\displaystyle G_{2}(a,b)$ $\displaystyle=\frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1)),$
 5.11.16 $\displaystyle H_{0}(a,b)$ $\displaystyle=1,$ $\displaystyle H_{1}(a,b)$ $\displaystyle=-\frac{1}{12}\binom{a-b}{2}(a-b+1),$ $\displaystyle H_{2}(a,b)$ $\displaystyle=\frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2}).$

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 $\displaystyle G_{k}(a,b)$ $\displaystyle=\binom{a-b}{k}\mathop{B^{(a-b+1)}_{k}\/}\nolimits\!\left(a\right),$ 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 $\displaystyle H_{k}(a,b)$ $\displaystyle=\binom{a-b}{2k}\mathop{B^{(a-b+1)}_{2k}\/}\nolimits\!\left(\frac% {a-b+1}{2}\right).$ 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 Referenced by: §5.11(iii) Permalink: http://dlmf.nist.gov/5.11.E18 Encodings: TeX, pMML, png

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

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

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

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