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.
Suppose there are two simple saddle points and of and suppose that they coalescent in the point .
We start with the cubic transformationof, 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
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.
Chester, Clive R.; Friedman, Bernard; Ursell, Fritz (1957). "An extension of the method of steepest descents". Mathematical Proceedings of the Cambridge Philosophical Society. 53 (3). Cambridge University Press: 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.
^Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 351. doi:10.1201/9781439864548.
^Chester, Clive R.; Friedman, Bernard; Ursell, Fritz (1957). "An extension of the method of steepest descents". Mathematical Proceedings of the Cambridge Philosophical Society. 53 (3). Cambridge University Press. doi:10.1017/S0305004100032655.