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So as to put our calculation of
in context, we first note that the expression (1.3) for has
been made more explicit by Jimbo et al. , who proved that
where satisfies the form of the Painlevé V equation:
subject to the boundary condition
Subsequent derivations of this result have been given by Its et al. ,
Mehta  and Tracy and Widom . Our expression for is given in
terms of the solution of a non-linear equation which generalizes (2.2).
We have obtained the following results. The p.d.f. for the
infinite GUE is given in
terms of a Fredholm determinant by
where is the integral operator on with kernel
( denotes the Bessel function) and b=1. Furthermore
(here the mean eigenvalue spacing is ), where satisfies
the non-linear equation
with b=1, subject to the boundary condition
with b=1 (the parameter b is included above for later convenience).
Note that with b=0 (2.7) reduces to (2.2).