6 INTRODUCTION The integral on the right is absolutely convergent and the end terms tend to zero as R -» oo. Furthermore if M is a bound for h and its first two derivatives then the above expressions can be estimated purely in terms of M. Thus if h depends on some auxiliary parameters and is uniformly bounded together with its first two derivatives in terms of these parameters then the integral J—co converges uniformly with respect to these parameters. (In particular, for Re A = 0 the integral f e~x^z"2f(z)dz is well defined.) Furthermore, we see that S e~Xt '2h{t)dt is a holomorphic function of A for Re A 0 and continuous for Re A 0, A ^ 0. In particular we can take h = 1. Now f^ e~u2/2du = ^/2TT and a change of variables for A real together with analytic continuation for A complex shows that 1/2 /. •-** * - (?r. for Re A 0, where the square root is computed by continuation from the positive real axis. Letting A -» Tik gives /.—/»*-(£)1/2 ±m/4 This completes the argument and establishes the formula. Assuming that j has only finitely many critical points in supp a, we can therefore assert that (S.P.) where H(y) is the Hessian of £ at y. Here sgn H denotes the signature of the quadratic form H. (For Q the signature is the number of +'s less the number of -'s , i.e., n — 21.) This formula is the method of stationary phase. We shall give a more invariant formulation of this result later on. We wish to apply this formula to our present circumstance. Observe that if f # 0 then d§ = 0 if and only if df2 = 0 and d2(&2) = td2j at each critical point. Since it is easier to deal with \\y — x\ than with \y — x\ we can use the above remark to simplify the calculations. It is pretty clear that for any fixed x the function $x(y) = \\y — x\ has a critical point if and only if the line joining x to y is orthogonal to the y surface. The Hessian of $x at any

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