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]

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 suppose 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

  • Chester, Clive R.; Friedman, Bernard; Ursell, Fritz (1957). "An extension of the method of steepest descents". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 53 (3): 604. doi:10.1017/S0305004100032655.
  • Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 351. doi:10.1201/9781439864548.
  • Wong, Roderick (2001). Asymptotic Approximations of Integrals. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898719260.
  • Temme, Nico M. (2014). Asymptotic Methods For Integrals. World Scientific. doi:10.1142/9195.

References

  1. Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 351. doi:10.1201/9781439864548.
  2. Chester, Clive R.; Friedman, Bernard; Ursell, Fritz (1957). "An extension of the method of steepest descents". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 53 (3). doi:10.1017/S0305004100032655.
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