# §8.22(i) Terminant Function

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

 8.22.1 $F_{p}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(p\right)}{2\pi}z^{1-p}\mathop% {E_{p}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(p% \right)}{2\pi}\mathop{\Gamma\/}\nolimits\!\left(1-p,z\right),$ Defines: $F_{p}(z)$: terminant function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $\mathop{E_{p}\/}\nolimits\!\left(z\right)$: generalized exponential integral, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$: incomplete gamma function, $z$: complex variable and $p$: parameter Permalink: http://dlmf.nist.gov/8.22.E1 Encodings: TeX, pMML, png

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 $\realpart{s}=\tfrac{1}{2}$. See Paris and Cang (1997).

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

 8.22.2 $\zeta_{x}(s)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{x}% \frac{t^{s-1}}{e^{t}-1}dt,$ $\realpart{s}>1$, Defines: $\zeta_{x}(s)$: incomplete Riemann zeta function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $dx$: differential of $x$, $e$: base of exponential function, $\int$: integral, $\realpart{}$: real part and $x$: real variable Permalink: http://dlmf.nist.gov/8.22.E2 Encodings: TeX, pMML, png

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

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

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

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