# §8.22 Mathematical Applications

## §8.22(i) Terminant Function

The so-called terminant function , 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

The function , with and , has an intimate connection with the Riemann zeta function 25.2(i)) on the critical line . See Paris and Cang (1997).

If denotes the incomplete Riemann zeta function defined by

8.22.2,

so that , then

For further information on , including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006).

The Debye functions and 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).