In asymptotic analysis, the method of Chester–Friedman–Ursell is a technique to find asymptotic expansions for contour integrals.
It was developed as an extension of the steepest descent method for getting uniform asymptotic expansions in the case of coalescing saddle points.
[1] The method was published in 1957 by Clive R. Chester, Bernard Friedman and Fritz Ursell.
[2] We study integrals of the form where
is a contour and Suppose we have two saddle points
with multiplicity
that depend on a parameter
exists, such that both saddle points coalescent to a new saddle point
, then the steepest descent method no longer gives uniform asymptotic expansions.
Suppose there are two simple saddle points
and suppose that they coalescent in the point
We start with the cubic transformation
, this means we introduce a new complex variable
and write where the coefficients
η := η ( α )
We have so the cubic transformation will be analytic and injective only if
must correspond to the zeros of
This gives the following system of equations we have to solve to determine
A theorem by Chester–Friedman–Ursell (see below) says now, that the cubic transform is analytic and injective in a local neighbourhood around the critical point
After the transformation the integral becomes where
and also at the coalescing point
Here ends the method and one can see the integral representation of the complex Airy function.
Chester–Friedman–Ursell note to write
not as a single power series but instead as to really get asymptotic expansions.
The cubic transformation with the above derived values for
η ( α )
corresponds to
, has only one branch point
in a local neighborhood of
the transformation is analytic and injective.