§8.18 Asymptotic Expansions of $I_x(a,b)$

Let

Then as $a\to \mathrm{\infty }$, with $b$ ($>0$) fixed,

uniformly for $x\in (0,1]$. The functions $F_k$ are defined by

with

and $Q(a,z)$ as in §8.2(i). The coefficients $d_k$ are defined by the generating function

In particular,

Compare also §24.16(i). A recurrence relation for the $d_k$ can be found in Nemes and Olde Daalhuis (2016).

Let

Then as $a+b\to \mathrm{\infty }$,

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

with $\eta /(x-x_0)>0$, and

with limiting value

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 $\eta =0$.

For the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ see (5.11.3).

Let $\mu =b/a$, and $x_0$ again be as in (8.18.8). Then as $a\to \mathrm{\infty }$

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

with $(\zeta -\mu )/(x_0-x)>0$, and

with limiting value

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 $x=x_0$).

For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation

see Temme (1992b).