  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*y, 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. It has been discovered that v(l') is a particular
solution of the ordinary differential equation (ODE) l'(1-l')d^2v/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 it
is mathematically impossible for simple Taylor expansion methods to 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.
