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8 Incomplete Gamma and Related FunctionsApplications

§8.22 Mathematical Applications

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§8.22(i) Terminant Function

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

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

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Keywords:
Debye functions, Riemann zeta function, incomplete Riemann zeta function, incomplete gamma functions
Referenced by:
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http://dlmf.nist.gov/8.22.ii
Addition (effective with 1.0.3):
The paragraph on Debye functions was added.

Reported 2011-08-21

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,
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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
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http://dlmf.nist.gov/8.22.E2
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so that \lim_{{x\to\infty}}\zeta_{x}(s)=\mathop{\zeta\/}\nolimits\!\left(s\right), then

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).

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