# §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)). $\mathop{\Gamma\/}\nolimits'\!\left(z\right)/\mathop{\Gamma\/}\nolimits\!\left(% z\right)$.

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

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

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