ABSTRACT
For the particular case of n = 2, we can parameterize the boundary by r: (z1(a),z2(1)) where 8 = 0 COrresponds to the maximum at X= Xo· The leading order term in (44.2) (e.g., the j = 0 term) can then be written as
) Af(Xo) 211" I J(l ,., e As I•"(O)I (Ho · N) • ...o' (44.3) if the maximum of 111 at 8 = 0 is simple (so that 111"(0) < 0). If ~t(O) is the curvature of rat x = Xo, then (44.3) can be further simplified to
44. Asymptotic Expansions: Multiple Integrals 201
Xo)A(x-Xo)T, where the matrix A = (Cli;) is defined by Cli; = t/>z,z1 (Jto). Let Q be an orthogonal matrix that diagonalizes A, i.e.,
QT AQ = ( ~1 • • . 0 ) . 0 ~"
Then define the variable z by (x-Xo) = QRzT where the matrix R = (ri;) is defined by ri; = 6i; l~il-l/'J.. Now the functions {~} are chosen so that (i = hi(z) = Zi + o(lzl) (as lzl - 0) and E~=l h1 = 2 (t/>(Xo) - t/>(x(z))).
