Definition of "asymptotics"

We shall see in sec. 6.2 that our result for S 3 ( a ) {\displaystyle S_{3}(a)} can be expressed in terms of special functions, whose asymptotics are well-known. There we show also that the asymptotic form for S 3 ( a ) {\displaystyle S_{3}(a)} obtained from the asymptotics of the special functions is identical to the asymptotic form obtained via Eq. (3.1).

1995 September 14, Vic Kowalenko, N. E. Frankel, L. Glasser, T. Taucher, Generalised Euler-Jacobi Inversion Formula and Asymptotics Beyond All Orders (Lecture note series / London mathematical society)‎, Cambridge University Press, page 15