With and , Li and Wong (2000) gives an asymptotic expansion for as , that holds uniformly for and in compact subintervals of . This expansion is in terms of the parabolic cylinder function and its derivative.
With and fixed, Qiu and Wong (2004) gives an asymptotic expansion for as , that holds uniformly for . This expansion is in terms of confluent hypergeometric functions. Asymptotic approximations are also provided for the zeros of in various cases depending on the values of and .
For two asymptotic expansions of as , with and fixed, see Jin and Wong (1998) and Wang and Wong (2011). The first expansion holds uniformly for , and the second for , being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions.
Dunster (2001b) provides various asymptotic expansions for as , in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for and for in unbounded intervals—each of which contains , where again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998).
For an asymptotic expansion of as , with fixed, see Li and Wong (2001). This expansion is uniformly valid in any compact -interval on the real line and is in terms of parabolic cylinder functions. Corresponding approximations are included for the zeros of .