# §8.22 Mathematical Applications

## §8.22(i) Terminant Function

The so-called terminant function $F_{p}\left(z\right)$, defined by

 8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$

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

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

 8.22.2 $\zeta_{x}(s)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{x}\frac{t^{s-1}}{e^{t}-1}% \,\mathrm{d}t,$ $\Re s>1$, ⓘ Defines: $\zeta_{x}(s)$: incomplete Riemann zeta function (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: 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), §8.22 and Ch.8

so that $\lim_{x\to\infty}\zeta_{x}(s)=\zeta\left(s\right)$, then

 8.22.3 $\zeta_{x}(s)=\sum_{k=1}^{\infty}k^{-s}P\left(s,kx\right),$ $\Re s>1$.

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

The Debye functions $\int_{0}^{x}t^{n}\left(e^{t}-1\right)^{-1}\,\mathrm{d}t$ and $\int_{x}^{\infty}t^{n}\left(e^{t}-1\right)^{-1}\,\mathrm{d}t$ 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).