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 expansion 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 where is a contour and are two analytic functions in the complex variable and continuous in . is a large number. Suppose we have two saddle points of with multiplicity that depend on a parameter . If now an exists, such that both saddle points coalescent to a new saddle point with multiplicity , then the steepest descent method no longer gives uniform asymptotic expansions. Procedure Suppose there are two simple saddle points and of and supposse that they coalescent in the point . We start with the cubic transformation of , this means we introduce a new complex variable and write where the coefficients and will be determined later. We have so the cubic transformation will be analytic and injective only if and are neither nor . Therefore and must correspond to the zeros of , i.e. with and . This gives the following system of equations we have to solve to determine and . 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 is the new contour for and The function is analytic at for and also at the coalescing point for . 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. Theorem by Chester–Friedman–Ursell Let and be as above. The cubic transformation with the above derived values for and , such that corresponds to , has only one branch point , so that for all in a local neighborhood of the transformation is analytic and injective. Literature References Asymptotic analysis