Technique to find asymptotic expansions
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. The method was published in 1957 by Clive R. Chester, Bernard Friedman and Fritz Ursell.
Method.
Setting.
We study integrals of the form
formula_1
where formula_2 is a contour and
Suppose we have two saddle points formula_7 of formula_8 with multiplicity formula_9 that depend on a parameter formula_5. If now an formula_11 exists, such that both saddle points coalescent to a new saddle point formula_12 with multiplicity formula_13, then the steepest descent method no longer gives uniform asymptotic expansions.
Procedure.
Suppose there are two simple saddle points formula_14 and formula_15 of formula_16 and suppose that they coalescent in the point formula_17.
We start with the "cubic transformation" formula_18 of formula_8, this means we introduce a new complex variable formula_20 and write
formula_21
where the coefficients formula_22 and formula_23 will be determined later.
We have
formula_24
so the cubic transformation will be analytic and injective only if formula_25 and formula_26 are neither formula_27 nor formula_28. Therefore formula_29 and formula_30 must correspond to the zeros of formula_31, i.e. with formula_32 and formula_33. This gives the following system of equations 
formula_34
we have to solve to determine formula_35 and formula_36. 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 formula_37.
After the transformation the integral becomes
formula_38
where formula_39 is the new contour for formula_20 and
formula_41
The function formula_42 is analytic at formula_43 for formula_44 and also at the coalescing point formula_45 for formula_11. Here ends the method and one can see the integral representation of the complex Airy function.
Chester–Friedman–Ursell note to write formula_42 not as a single power series but instead as
formula_48
to really get asymptotic expansions.
Theorem by Chester–Friedman–Ursell.
Let formula_49 and formula_50 be as above. The cubic transformation
formula_51
with the above derived values for formula_52 and formula_53, such that formula_54 corresponds to formula_55, has only one branch point formula_56, so that for all formula_5 in a local neighborhood of formula_11 the transformation is analytic and injective.