ar X iv : 0 70 8 . 09 96 v 2 [ m at h - ph ] 8 A ug 2 00 7 Formal derivation of an exact series expansion for the Principal Field Emission Elliptic Function v

An exact series expansion is now known for the Principal Field Emission Elliptic Function v, in terms of a complementary elliptic variable l ′ equal to y 2 , where y is the Nordheim parameter. This expansion was originally found by using the algebraic manipulation package MAPLE. This paper presents a formal mathematical derivation. Recently, it has been discovered that v(l ′) is a particular solution of the ordinary differential equation (ODE) l ′ (1−l ′)d 2 v/dl ′2 = nv, when the index n = 3/16. This ODE appears to be new in mathematical physics and elliptic-function theory. The paper first uses an 1876 result from Cayley to establish the boundary condition that dv/dl ′ satisfies as l ′ tends to zero. It then uses the method of Frobenius to obtain two linearly independent series solutions for the ODE, and hence derives the series expansion for v(l ′). It is shown that terms in ln l ′ are required in a mathematically correct solution, but fractional powers of l ′ are not. The form of the ODE also implies that simple Taylor expansion methods cannot generate good approximation formulae valid over the whole range 0 ≤ l ′ ≤ 1; this conclusion may also apply to barriers of other shapes. It is hoped that this derivation might serve as a paradigm for the treatment of other tunnelling barrier models for cold field electron emission, if in any particular case an ODE can be found for which the tunnelling-exponent correction function is a particular solution.