§8.1 Special Notation

Keywords:: Prym’s functions, generalized exponential integral, generalized sine and cosine integrals, incomplete beta functions, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.1

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

$x$	real variable.
$z$	complex variable.
$a,p$	real or complex parameters.
$k,n$	nonnegative integers.
$\delta$	arbitrary small positive constant.
$\mathop{\Gamma\/}\nolimits\!\left(z\right)$	gamma function (§5.2(i)).
$\mathop{\psi\/}\nolimits\!\left(z\right)$	${\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(z\right)/\mathop{\Gamma\/}\nolimits\!\left(z\right)$ .

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).