# §8.1 Special Notation

(For other notation see Notation for the Special Functions.)

$x$ real variable. complex variable. real or complex parameters. nonnegative integers. arbitrary small positive constant. gamma function (§5.2(i)). $\Gamma'\left(z\right)/\Gamma\left(z\right)$.

Unless otherwise indicated, primes denote derivatives with respect to the argument.

The functions treated in this chapter are the incomplete gamma functions $\gamma\left(a,z\right)$, $\Gamma\left(a,z\right)$, $\gamma^{*}\left(a,z\right)$, $P\left(a,z\right)$, and $Q\left(a,z\right)$; the incomplete beta functions $\mathrm{B}_{x}\left(a,b\right)$ and $I_{x}\left(a,b\right)$; the generalized exponential integral $E_{p}\left(z\right)$; the generalized sine and cosine integrals $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$.

Alternative notations include: Prym’s functions $P_{z}(a)=\gamma\left(a,z\right)$, $Q_{z}(a)=\Gamma\left(a,z\right)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma\left(a+1,z\right)$, $[a,z]!=\Gamma\left(a+1,z\right)$, Dingle (1973); $B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)$, $I(a,b,x)=I_{x}\left(a,b\right)$, Magnus et al. (1966); $\operatorname{Si}\left(a,x\right)\to\operatorname{Si}\left(1-a,x\right)$, $\operatorname{Ci}\left(a,x\right)\to\operatorname{Ci}\left(1-a,x\right)$, Luke (1975).