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8 Incomplete Gamma and Related FunctionsApplications

§8.22 Mathematical Applications


§8.22(i) Terminant Function

The so-called terminant function Fp(z), defined by

8.22.1 Fp(z)=Γ(p)2πz1-pEp(z)=Γ(p)2πΓ(1-p,z),

plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. See §§2.11(ii)2.11(v) and the references supplied in these subsections.

§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

The function Γ(a,z), with |pha|12π and phz=12π, has an intimate connection with the Riemann zeta function ζ(s)25.2(i)) on the critical line s=12. See Paris and Cang (1997).

If ζx(s) denotes the incomplete Riemann zeta function defined by

8.22.2 ζx(s)=1Γ(s)0xts-1et-1dt,

so that limxζx(s)=ζ(s), then

8.22.3 ζx(s)=k=1k-sP(s,kx),

For further information on ζx(s), including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006).

The Debye functions 0xtn(et-1)-1dt and xtn(et-1)-1dt are closely related to the incomplete Riemann zeta function and the Riemann zeta function. See Abramowitz and Stegun (1964, p. 998) and Ashcroft and Mermin (1976, Chapter 23).