8 Incomplete Gamma and Related Functions Related Functions 8.17 Incomplete Beta Functions 8.19 Generalized Exponential Integral

§8.18 Asymptotic Expansions of $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$

Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/8.18

§8.18(i) Large Parameters, Fixed
§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

§8.18(i) Large Parameters, Fixed

Notes:: Use (8.17.9) and apply §15.12(ii).
Permalink:: http://dlmf.nist.gov/8.18.i

If and are fixed, with and , then as $a\to\infty$

8.18.1 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)={\mathop{\Gamma\/}\nolimits\!% \left(a+b\right)x^{a}(1-x)^{{b-1}}}\*\left(\sum_{{k=0}}^{{n-1}}\frac{1}{% \mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)\mathop{\Gamma\/}\nolimits\!% \left(b-k\right)}\left(\frac{x}{1-x}\right)^{k}+\mathop{O\/}\nolimits\!\left(% \frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a+n+1\right)}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : nonnegative integer, : nonnegative integer, : parameter, : parameter and : variable
Referenced by:: §8.18(i)
Permalink:: http://dlmf.nist.gov/8.18.E1
Encodings:: TeX, pMML, png

for each $n=0,1,2,\dots$ . If $b=1,2,3,\dots$ and $n\geq b$ , then the $\mathop{O\/}\nolimits$ -term can be omitted and the result is exact.

If $b\to\infty$ and and are fixed, with and , then (8.18.1), with and interchanged and replaced by , can be combined with (8.17.4).

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Notes:: See Temme (1996b, §§11.3.3.1–11.3.3.3).
Permalink:: http://dlmf.nist.gov/8.18.ii

¶ Large , Fixed

Let

8.18.2 $\xi=-\mathop{\ln\/}\nolimits x.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : variable and $\xi$ : change of variable
Permalink:: http://dlmf.nist.gov/8.18.E2
Encodings:: TeX, pMML, png

Then as $a\to\infty$ , with () fixed,

8.18.3 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)\sim\frac{\mathop{\Gamma\/}% \nolimits\!\left(a+b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\sum_{% {k=0}}^{\infty}d_{k}F_{k},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, $\sim$ : asymptotic equality, : nonnegative integer, : parameter, : parameter, : variable, $F_{k}$ : functions and $d_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E3
Encodings:: TeX, pMML, png

uniformly for $x\in(0,1)$ . The functions $F_{k}$ are defined by

8.18.4 $aF_{{k+1}}=(k+b-a\xi)F_{k}+k\xi F_{{k-1}},$

Symbols:: : nonnegative integer, : parameter, : parameter, $\xi$ : change of variable and $F_{k}$ : functions
Permalink:: http://dlmf.nist.gov/8.18.E4
Encodings:: TeX, pMML, png

with

8.18.5

$F_{0}=a^{{-b}}\mathop{Q\/}\nolimits\!\left(b,a\xi\right),$

$F_{1}=\frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{{-a\xi}}}{a\mathop{\Gamma\/}% \nolimits\!\left(b\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, : parameter, : parameter, $\xi$ : change of variable and $F_{k}$ : functions
Permalink:: http://dlmf.nist.gov/8.18.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

and $\mathop{Q\/}\nolimits\!\left(a,z\right)$ as in §8.2(i). The coefficients $d_{k}$ are defined by the generating function

8.18.6 $\left(\frac{1-e^{{-t}}}{t}\right)^{{b-1}}=\sum_{{k=0}}^{\infty}d_{k}(t-\xi)^{k}.$

Symbols:: : base of exponential function, : nonnegative integer, : parameter, $\xi$ : change of variable and $d_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E6
Encodings:: TeX, pMML, png

In particular,

8.18.7

$d_{0}=\left(\frac{1-x}{\xi}\right)^{{b-1}},$

$d_{1}=\frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}.$

Symbols:: : parameter, : variable, $\xi$ : change of variable and $d_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

Compare also §24.16(i).

¶ Symmetric Case

Keywords:: incomplete beta functions

Let

8.18.8 $x_{0}=a/(a+b).$

Symbols:: : parameter, : parameter and : variable
Referenced by:: ¶ ‣ §8.18(ii)
Permalink:: http://dlmf.nist.gov/8.18.E8
Encodings:: TeX, pMML, png

Then as $a+b\to\infty$ ,

8.18.9 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)\sim\tfrac{1}{2}\mathop{\mathrm{% erfc}\/}\nolimits\!\left(-\eta\sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi(a+b)}}\*% \left(\frac{x}{x_{0}}\right)^{a}\left(\frac{1-x}{1-x_{0}}\right)^{b}\sum_{{k=0% }}^{\infty}\frac{(-1)^{k}c_{k}(\eta)}{(a+b)^{k}},$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\sim$ : asymptotic equality, : nonnegative integer, : parameter, : parameter, : variable and $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E9
Encodings:: TeX, pMML, png

uniformly for $x\in(0,1)$ and , $b/(a+b)\in[\delta,1-\delta]$ , where $\delta$ again denotes an arbitrary small positive constant. For $\mathop{\mathrm{erfc}\/}\nolimits$ see §7.2(i). Also,

8.18.10 $-\tfrac{1}{2}\eta^{2}=x_{0}\mathop{\ln\/}\nolimits\!\left(\frac{x}{x_{0}}% \right)+(1-x_{0})\mathop{\ln\/}\nolimits\!\left(\frac{1-x}{1-x_{0}}\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : variable
Permalink:: http://dlmf.nist.gov/8.18.E10
Encodings:: TeX, pMML, png

with $\eta/(x-x_{0})>0$ , and

8.18.11 $c_{0}(\eta)=\frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}},$

Symbols:: : variable and $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E11
Encodings:: TeX, pMML, png

with limiting value

8.18.12 $c_{0}(0)=\frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}.$

Symbols:: : variable and $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E12
Encodings:: TeX, pMML, png

For this result, and for higher coefficients $c_{k}(\eta)$ see Temme (1996b, §11.3.3.2). All of the $c_{k}(\eta)$ are analytic at $\eta=0$ .

¶ General Case

Keywords:: gamma function, incomplete beta functions, scaled gamma function

Let $\widetilde{\Gamma}(z)$ denote the scaled gamma function

8.18.13 $\widetilde{\Gamma}(z)=(2\pi)^{{-1/2}}e^{z}z^{{(1/2)-z}}\mathop{\Gamma\/}% \nolimits\!\left(z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function and : complex variable
Permalink:: http://dlmf.nist.gov/8.18.E13
Encodings:: TeX, pMML, png

$\mu=b/a$ , and $x_{0}$ again be as in (8.18.8). Then as $a\to\infty$

8.18.14 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)\sim\mathop{Q\/}\nolimits\!\left(% b,a\zeta\right)-\frac{(2\pi b)^{{-1/2}}}{\widetilde{\Gamma}(b)}\left(\frac{x}{% x_{0}}\right)^{a}\left(\frac{1-x}{1-x_{0}}\right)^{b}\sum_{{k=0}}^{\infty}% \frac{h_{k}(\zeta,\mu)}{a^{k}},$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\sim$ : asymptotic equality, : nonnegative integer, : parameter, : parameter, : variable, $\mu$ , $\zeta$ : variable and $h_{k}(\zeta,\mu)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E14
Encodings:: TeX, pMML, png

uniformly for $b\in(0,\infty)$ and $x\in(0,1)$ . Here

8.18.15 $\mu\mathop{\ln\/}\nolimits\zeta-\zeta=\mathop{\ln\/}\nolimits x+\mu\mathop{\ln% \/}\nolimits\!\left(1-x\right)+(1+\mu)\mathop{\ln\/}\nolimits\!\left(1+\mu% \right)-\mu,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : variable, $\mu$ and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/8.18.E15
Encodings:: TeX, pMML, png

with $(\zeta-\mu)/(x_{0}-x)>0$ , and

8.18.16 $h_{0}(\zeta,\mu)=\mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{{-3/2}}}{x_{0}-x}% \right),$

Symbols:: : variable, $\mu$ , $\zeta$ : variable and $h_{k}(\zeta,\mu)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E16
Encodings:: TeX, pMML, png

with limiting value

8.18.17 $h_{0}(\mu,\mu)=\frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right).$

Symbols:: $\mu$ and $h_{k}(\zeta,\mu)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E17
Encodings:: TeX, pMML, png

For this result and higher coefficients $h_{k}(\zeta,\mu)$ see Temme (1996b, §11.3.3.3). All of the $h_{k}(\zeta,\mu)$ are analytic at $\zeta=\mu$ (corresponding to $x=x_{0}$ ).