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8 Incomplete Gamma and Related FunctionsNotation

§8.1 Special Notation

(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.
δ arbitrary small positive constant.
Γ(z) gamma function (§5.2(i)).
ψ(z) Γ(z)/Γ(z).

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

The functions treated in this chapter are the incomplete gamma functions γ(a,z), Γ(a,z), γ(a,z), P(a,z), and Q(a,z); the incomplete beta functions Bx(a,b) and Ix(a,b); the generalized exponential integral Ep(z); the generalized sine and cosine integrals si(a,z), ci(a,z), Si(a,z), and Ci(a,z).

Alternative notations include: Prym’s functions Pz(a)=γ(a,z), Qz(a)=Γ(a,z), Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); (a,z)!=γ(a+1,z), [a,z]!=Γ(a+1,z), Dingle (1973); B(a,b,x)=Bx(a,b), I(a,b,x)=Ix(a,b), Magnus et al. (1966); Si(a,x)Si(1a,x), Ci(a,x)Ci(1a,x), Luke (1975).