§8.22 Mathematical Applications

The so-called terminant function $F_p\left(z\right)$, defined by

8.22.1		$$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$
Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter Permalink: http://dlmf.nist.gov/8.22.E1 Encodings: TeX, pMML, png See also: Annotations for 8.22(i)

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.

The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. See Paris and Cang (1997).

If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by

8.22.2		$${\zeta }_x(s)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^x\frac{t^{s-1}}{{\mathrm{e}}^t-1}dt,$$
$\mathrm{\Re }s>1$,
Defines: ${\zeta }_x(s)$: incomplete Riemann zeta function (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/8.22.E2 Encodings: TeX, pMML, png See also: Annotations for 8.22(ii)

so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then

8.22.3		$${\zeta }_x(s)=\sum _{k=1}^{\mathrm{\infty }}{k}^{-s}P(s,kx),$$
$\mathrm{\Re }s>1$.
Symbols: $P(a,z)$: normalized incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $k$: nonnegative integer and ${\zeta }_x(s)$: incomplete Riemann zeta function Permalink: http://dlmf.nist.gov/8.22.E3 Encodings: TeX, pMML, png See also: Annotations for 8.22(ii)

For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006).

The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ 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).