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