A historical note is necessary. The idea that the Schottky–Nordheim barrier needed a correction factor, as in eq. (9), was introduced by Nordheim in 1928, but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, Kroemer and Houston in 1953, and its mathematics was developed further by Murphy and Good in 1956. This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable y known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ℓ′ (= y). And only recently has it been possible to complete the definition of ν (ℓ′) by developing and proving the validity of an exact series expansion for this function (by starting from known special-case solutions of the Gauss hypergeometric differential equation). Also, approximation (11) has been found only recently. Approximation (11) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course.