American Journal of Mathematics, Vol. 50, No. 2 (April 1928), 269-302

Introduction

In the present paper, the author attempts to give unity to the investigations of the asymptotic expansions for large values of n, of functions defined by the integral

I = ∫ e^{nf(z)}F(z) dz

taken along paths in the z-plane. The problem of grouping paths in such a way that the asymptotic expansions of the functions, defined by the integral I along paths belonging to the same group are identical, is first considered. The asymptotic evaluation of the integral I, along any one of a large class of paths is then effected by means of the discussion of the value of the integral I along certain standard paths.

This procedure is precisely similar to that adopted in the theory of complex integration. Cauchy's theorem gives specifications of a group of paths associated with two points, such that the value of the integral

∫_{C} F(z) dz

is the same, whichever path of the group C may be. Next, the value of the integral taken along any one of a wide class of paths may be obtained by the discussion of the value of the integral taken along certain standard paths. The calculus of residues belongs to this part of the subject.

In Cauchy's theorem, the paths belonging to one group, all begin at one point and end at a second point. In the corresponding theorem of asymptotic evaluation, it is sufficient to specify a certain *continuum* in which the paths belonging to one group must start and a certain continuum in which they must end. The continua in question are of a special type; they are called, in this paper, 'valleys' or 'valley continua.' An 'M-valley' associated with the function f(z) is a continuum in which the real part of f(z) has M as its upper bound. Under certain circumstances, if m_{1} and m_{2} are two M-valleys associated with the function f(z), and if C is a path starting in the continuum m_{1} and ending in the continuum m_{2}, the asymptotic value of the integral

e^{–nM} ∫ e^{nf(z)} F(z) dz

is the same whichever path C may be. The results are obtained by considering the deductions which can be drawn from the behaviour of the real part of the function f(z) on the path C, as to the asymptotic value, for large values of n, of the integral I taken along the path C. Although, on occasion it has been pointed out that the integral I taken along a certain path differs from the integral taken along a second path C' by a term of order e^{–nδ}, where δ is some positive number, the author is unaware of any previous attempt to study the general problem of grouping paths in such a way, that the functions defined by the integral I taken along paths belonging to one group admit the same asymptotic expansion for large values of n.