# §8.22 Mathematical Applications

## §8.22(i) Terminant Function

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

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

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 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$, with $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\tfrac{1}{2}\pi$ and $\mathop{\mathrm{ph}\/}\nolimits z=\tfrac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\mathop{\zeta\/}\nolimits\!\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 $\zeta_{x}(s)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{x}% \frac{t^{s-1}}{e^{t}-1}\mathrm{d}t,$ $\Re{s}>1$, Defines: $\zeta_{x}(s)$: incomplete Riemann zeta function (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: 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)

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

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

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).