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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. [4], 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. [5], Mehta [6] and Tracy and Widom [7]. 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

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