About the Project
36 Integrals with Coalescing SaddlesApplications

§36.12 Uniform Approximation of Integrals

  1. §36.12(i) General Theory for Cuspoids
  2. §36.12(ii) Special Case
  3. §36.12(iii) Additional References

§36.12(i) General Theory for Cuspoids

The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. In the cuspoid case (one integration variable)

36.12.1 I(𝐲,k)=exp(ikf(u;𝐲))g(u,𝐲)du,

where k is a large real parameter and 𝐲={y1,y2,} is a set of additional (nonasymptotic) parameters. As 𝐲 varies as many as K+1 (real or complex) critical points of the smooth phase function f can coalesce in clusters of two or more. The function g has a smooth amplitude. Also, f is real analytic, and K+2f/uK+2>0 for all 𝐲 such that all K+1 critical points coincide. If K+2f/uK+2<0, then we may evaluate the complex conjugate of I for real values of 𝐲 and g, and obtain I by conjugation and analytic continuation. The critical points uj(𝐲), 1jK+1, are defined by

36.12.2 uf(uj(𝐲);𝐲)=0.

The leading-order uniform asymptotic approximation is given by

where A(𝐲), 𝐳(𝐲,k), am(𝐲) are as follows. Define a mapping u(t;𝐲) by relating f(u;𝐲) to the normal form (36.2.1) of ΦK(t;𝐱) in the following way:

36.12.4 f(u(t,𝐲);𝐲)=A(𝐲)+ΦK(t;𝐱(𝐲)),

with the K+1 functions A(𝐲) and 𝐱(𝐲) determined by correspondence of the K+1 critical points of f and ΦK. Then

36.12.5 f(uj(𝐲);𝐲)=A(𝐲)+ΦK(tj(𝐱(𝐲));𝐱(𝐲)),

where tj(𝐱), 1jK+1, are the critical points of ΦK, that is, the solutions (real and complex) of (36.4.1). Correspondence between the uj(𝐲) and the tj(𝐱) is established by the order of critical points along the real axis when 𝐲 and 𝐱 are such that these critical points are all real, and by continuation when some or all of the critical points are complex. The branch for 𝐱(𝐲) is such that 𝐱 is real when 𝐲 is real. In consequence,

36.12.6 A(𝐲)=f(u(0,𝐲);𝐲),
36.12.7 𝐳(𝐲;k) ={z1(𝐲;k),z2(𝐲;k),,zK(𝐲;k)},
zm(𝐲;k) =xm(𝐲)k1(m/(K+2)),
36.12.8 am(𝐲)=n=1K+1Pmn(𝐲)Gn(𝐲)(tn(𝐱(𝐲)))m+1l=1lnK+1(tn(𝐱(𝐲))tl(𝐱(𝐲))),


36.12.9 Pmn(𝐲)=(tn(𝐱(𝐲)))K+1+l=m+2KlK+2xl(𝐲)(tn(𝐱(𝐲)))l1,


36.12.10 Gn(𝐲)=g(tn(𝐲),𝐲)2ΦK(tn(𝐱(𝐲));𝐱(𝐲))/t22f(un(𝐲))/u2.

In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. The square roots are real and positive when 𝐲 is such that all the critical points are real, and are defined by analytic continuation elsewhere. The quantities am(𝐲) are real for real 𝐲 when g is real analytic.

This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ΨK(𝐱;k) in (36.2.10) away from 𝐱=𝟎, in terms of canonical integrals ΨJ(ξ(𝐱;k)) for J<K. For example, the diffraction catastrophe Ψ2(x,y;k) defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function Ψ1(ξ(x,y;k)) when k is large, provided that x and y are not small. For details of this example, see Paris (1991).

For further information see Berry and Howls (1993).

§36.12(ii) Special Case

For K=1, with a single parameter y, let the two critical points of f(u;y) be denoted by u±(y), with u+>u for those values of y for which these critical points are real. Then

36.12.11 I(y,k)=Δ1/4π2k1/3exp(ikf~)×((g+f+′′+gf′′)Ai(k2/3Δ)(1+O(1k))i(g+f+′′gf′′)Ai(k2/3Δ)k1/3Δ1/2(1+O(1k))),


36.12.12 f~ =12(f(u+(y),y)+f(u(y),y)),
g± =g(u±(y),y),
f±′′ =2u2f(u±(y),y),
Δ =(34(f(u(y),y)f(u+(y),y)))2/3.

For Ai and Ai see §9.2. Branches are chosen so that Δ is real and positive if the critical points are real, or real and negative if they are complex. The coefficients of Ai and Ai are real if y is real and g is real analytic. Also, Δ1/4/f+′′ and Δ1/4/f′′ are chosen to be positive real when y is such that both critical points are real, and by analytic continuation otherwise.

§36.12(iii) Additional References

For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).