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36 Integrals with Coalescing SaddlesProperties

§36.2 Catastrophes and Canonical Integrals


§36.2(i) Definitions

Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K

36.2.1 ΦK(t;x)=tK+2+m=1Kxmtm.

Special cases: K=1, fold catastrophe; K=2, cusp catastrophe; K=3, swallowtail catastrophe.

Normal Forms for Umbilic Catastrophes with Codimension K=3

36.2.2 Φ(E)(s,t;x) =s3-3st2+z(s2+t2)+yt+xs,
(elliptic umbilic).
36.2.3 Φ(H)(s,t;x) =s3+t3+zst+yt+xs,
(hyperbolic umbilic).

Canonical Integrals

36.2.4 ΨK(x)=-exp(ΦK(t;x))t.
36.2.5 Ψ(U)(x)=--exp(Φ(U)(s,t;x))st,
36.2.6 Ψ(E)(x)=2π/3exp((427z3+13xz-14π))×exp(-7π/12)exp(π/12)exp((u6+2zu4+(z2+x)u2+y212u2))u,

with the contour passing to the lower right of u=0.

36.2.7 Ψ(E)(x) =4π31/3exp((227z3-13xz))(exp(-π6)F+(x)+exp(π6)F-(x)),
F±(x) =0cos(ryexp(±π6))exp(2r2zexp(±π3))×Ai(32/3r2+3-1/3exp(π3)(13z2-x))r.
36.2.8 Ψ(H)(x)=4π/6exp((127z3+16z(y+x)+14π))×exp(5π/12)exp(π/12)exp((2u6+2zu4+(12z2+x+y)u2-(y-x)224u2))u,

with the contour passing to the upper right of u=0.

36.2.9 Ψ(H)(x)=2π31/3exp(5π/6)exp(π/6)exp((s3+xs))Ai(zs+y31/3)s.

Diffraction Catastrophes

36.2.10 ΨK(x;k)=k-exp(kΦK(t;x))t,
36.2.11 Ψ(U)(x;k)=k--exp(kΦ(U)(s,t;x))st,
U=E,H; k>0.

For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975).

§36.2(ii) Special Cases

36.2.12 Ψ0=πexp(π4).

Ψ1 is related to the Airy function (§9.2):

36.2.13 Ψ1(x)=2π31/3Ai(x31/3).

Ψ2 is the Pearcey integral (Pearcey (1946)):

36.2.14 Ψ2(x)=P(x2,x1)=-exp((t4+x2t2+x1t))t.

(Other notations also appear in the literature.)

36.2.15 ΨK(0)=2K+2Γ(1K+2){exp(π2(K+2)),K even,cos(π2(K+2)),K odd.
36.2.16 Ψ1(0) =1.54669,
Ψ2(0) =1.67481+ 0.69373
Ψ3(0) =1.74646,
Ψ4(0) =1.79222+ 0.48022.
36.2.17 px1pΨK(0) =2K+2Γ(p+1K+2)cos(π2(p+1K+2+p)),
K odd,
2q+1x12q+1ΨK(0) =0,
K even,
2qx12qΨK(0) =2K+2Γ(2q+1K+2)exp(π2(2q+1K+2+2q)),
K even.
36.2.18 Ψ(E)(0) =13πΓ(16)
Ψ(H)(0) =13Γ2(13)
36.2.19 Ψ2(0,y)=π2|y|2exp(-y28)(exp(π8)J-1/4(y28)-sign(y)exp(-π8)J1/4(y28)).

For the Bessel function J see §10.2(ii).

36.2.20 Ψ(E)(x,y,0)=2π2(23)2/3(Ai(x+y121/3)Bi(x-y121/3)),
36.2.21 Ψ(H)(x,y,0)=4π232/3Ai(x31/3)Ai(y31/3).

Addendum: For further special cases see §36.2(iv)

§36.2(iii) Symmetries

36.2.22 Ψ2K(x)=Ψ2K(x),
x2m+1=-x2m+1, x2m=x2m.
36.2.23 Ψ2K+1(x)=Ψ2K+1(x),
x2m+1=x2m+1, x2m=-x2m.
36.2.24 Ψ(U)(x,y,z)=Ψ(U)(x,y,-z),
36.2.25 Ψ(E)(x,-y,z)=Ψ(E)(x,y,z).
36.2.26 Ψ(E)(-12x32y,±32x-12y,z)=Ψ(E)(x,y,z),

(rotation by ±23π in x,y plane).

36.2.27 Ψ(H)(x,y,z)=Ψ(H)(y,x,z).

§36.2(iv) Addendum to 36.2(ii) Special Cases

36.2.28 Ψ(E)(0,0,z)=Ψ(E)(0,0,-z)=2ππz27exp(227z3)(J-1/6(227z3)+J1/6(227z3)),
36.2.29 Ψ(H)(0,0,z)=Ψ(H)(0,0,-z)=21/33exp(127z3)Ψ(E)(0,0,-z22/3),

Here the functions Ψ(E) and Ψ(H) are the complex conjugates of the functions Ψ(E) and Ψ(H), respectively.