Define

where the branch of the square root is continuous and satisfies $\eta (\lambda )\sim \lambda -1$ as $\lambda \to 1$. Then

Also, denote

and

where $F\left(x\right)$ is Dawson’s integral; see §7.2(ii). Then as $a\to \mathrm{\infty }$ in the sector ,

in each case uniformly with respect to $\lambda$ in the sector $|\mathrm{ph}\lambda |\le 2\pi -\delta$ ().

With $\mu =\lambda -1$, the coefficients $c_k(\eta )$ are given by

where $g_k$, $k=0,1,2,\mathrm{\dots }$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\mathrm{\Gamma }\left(z\right)$. The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by

where $d_{0,0}=-\frac{1}{3}$,

and ${\alpha }_3,{\alpha }_4,\mathrm{\dots }$ are defined by

In particular,

For numerical values of $d_{k,n}$ to 30D for $k=0(1)9$ and $n=0(1)N_k$, where $N_k=28-4\lfloor k/2\rfloor$, see DiDonato and Morris (1986).

Special cases are given by

where

For error bounds for (8.12.7) see Paris (2002a). For the asymptotic behavior of $c_k(\eta )$ as $k\to \mathrm{\infty }$ see Dunster et al. (1998) and Olde Daalhuis (1998c). The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for $Q(a,z)$.

A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by

for $z\to \mathrm{\infty }$ in , with $\mathrm{\Re }\left(z-a\right)\le 0$ for $P(a,z)$ and $\mathrm{\Re }\left(z-a\right)\ge 0$ for $Q(a,z)$. Here

and

Higher coefficients $A_k(\chi )$, $B_k(\chi )$, up to $k=8$, are given in Paris (2002b).

Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a).

For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a).