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36 Integrals with Coalescing SaddlesProperties

§36.2 Catastrophes and Canonical Integrals

Contents
  1. §36.2(i) Definitions
  2. §36.2(ii) Special Cases
  3. §36.2(iii) Symmetries
  4. §36.2(iv) Addendum to 36.2(ii) Special Cases

§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) =s33st2+z(s2+t2)+yt+xs,
x={x,y,z},
(elliptic umbilic).
36.2.3 Φ(H)(s,t;x) =s3+t3+zst+yt+xs,
x={x,y,z},
(hyperbolic umbilic).

Canonical Integrals

36.2.4 ΨK(x)=exp(iΦK(t;x))dt.
36.2.5 Ψ(U)(x)=exp(iΦ(U)(s,t;x))dsdt,
U=E,H.
36.2.6 Ψ(E)(x)=2π/3exp(i(427z3+13xz14π))×exp(7πi/12)exp(πi/12)exp(i(u6+2zu4+(z2+x)u2+y212u2))du,

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

36.2.7 Ψ(E)(x) =4π31/3exp(i(227z313xz))(exp(iπ6)F+(x)+exp(iπ6)F(x)),
F±(x) =0cos(ryexp(±iπ6))exp(2ir2zexp(±iπ3))×Ai(32/3r2+31/3exp(iπ3)(13z2x))dr.
36.2.8 Ψ(H)(x)=4π/6exp(i(127z3+16z(y+x)+14π))×exp(5πi/12)exp(πi/12)exp(i(2u6+2zu4+(12z2+x+y)u2(yx)224u2))du,

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

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

Diffraction Catastrophes

36.2.10 ΨK(x;k)=kexp(ikΦK(t;x))dt,
k>0.
36.2.11 Ψ(U)(x;k)=kexp(ikΦ(U)(s,t;x))dsdt,
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(iπ4).

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(i(t4+x2t2+x1t))dt.

(Other notations also appear in the literature.)

36.2.15 ΨK(0)=2K+2Γ(1K+2){exp(iπ2(K+2)),K even,cos(π2(K+2)),K odd.
36.2.16 Ψ1(0) =1.54669,
Ψ2(0) =1.67481+i 0.69373
Ψ3(0) =1.74646,
Ψ4(0) =1.79222+i 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(iπ2(2q+1K+2+2q)),
K even.
36.2.18 Ψ(E)(0) =13πΓ(16)=3.28868,
Ψ(H)(0) =13Γ2(13)=2.39224.
36.2.19 Ψ2(0,y)=π2|y|2exp(iy28)(exp(iπ8)J1/4(y28)sign(y)exp(iπ8)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+iy121/3)Bi(xiy121/3)),
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)¯,
U=E,H.
36.2.25 Ψ(E)(x,y,z)=Ψ(E)(x,y,z).
36.2.26 Ψ(E)(12x32y,±32x12y,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(227iz3)(J1/6(227z3)+iJ1/6(227z3)),
z0,
36.2.29 Ψ(H)(0,0,z)=Ψ(H)(0,0,z)¯=21/33exp(127iz3)Ψ(E)(0,0,z22/3),
<z<.