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36 Integrals with Coalescing SaddlesProperties

§36.7 Zeros

  1. §36.7(i) Fold Canonical Integral
  2. §36.7(ii) Cusp Canonical Integral
  3. §36.7(iii) Elliptic Umbilic Canonical Integral
  4. §36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

§36.7(i) Fold Canonical Integral

This is the Airy function Ai9.2).

§36.7(ii) Cusp Canonical Integral

This is (36.2.4) and (36.2.1) with K=2.

The zeros in Table 36.7.1 are points in the 𝐱=(x,y) plane, where phΨ2(𝐱) is undetermined. All zeros have y<0, and fall into two classes. Inside the cusp, that is, for x2<8|y|3/27, the zeros form pairs lying in curved rows. Close to the y-axis the approximate location of these zeros is given by

36.7.1 ym =2π(2m+1),
xm,n± =2ym(2n+12+(1)m12±14)π,
m=1,2,3,, n=0,±1,±2,.
Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D.
Zeros {xy} inside, and zeros [xy] outside, the cusp x2=827|y|3.
{±0.527684.37804} [±2.352181.74360]
{±1.411015.55470} {±2.360945.52321} [±4.427073.05791]
{±0.430396.64285} {±3.063896.44624} {±3.958066.40312} [±6.161854.03551]
{±1.216057.49906} {±2.029227.48629} {±4.565377.19629} {±5.422067.14718} [±7.723524.84817]
{±0.384888.31916} {±2.711938.22315} {±3.492868.20326} {±5.966697.85723} {±6.795387.80456} [±9.173085.55831]

More general asymptotic formulas are given in Kaminski and Paris (1999). Just outside the cusp, that is, for x2>8|y|3/27, there is a single row of zeros on each side. With n=0,1,2,, they are located approximately at

36.7.2 xn =±(827)1/2|yn|3/2(1+ξn),
yn =(3π(8n+5)9+8ξn)1/2,

where ξn is the real solution of

36.7.3 3π(8n+5)9+8ξnξn3/2=2716(32)1/2(ln(1ξn)+3ln(32)).

For a more extensive asymptotic analysis and further tabulations, see Kaminski and Paris (1999).

§36.7(iii) Elliptic Umbilic Canonical Integral

This is (36.2.5) with (36.2.2). The zeros are lines in 𝐱=(x,y,z) space where phΨ(E)(𝐱) is undetermined. Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the z-axis that is far from the origin, the zero contours form an array of rings close to the planes

36.7.4 zn=±3(14π(2n12))1/3=3.48734(n14)1/3,

Near z=zn, and for small x and y, the modulus |Ψ(E)(𝐱)| has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by

36.7.5 Δz =9π2zn2,
Δx =6πzn.

The zeros are approximated by solutions of the equation

36.7.6 exp(2πi(zznΔz+2xΔx))(2exp(6πixΔx)cos(23πyΔx)+1)=3.

The rings are almost circular (radii close to (Δx)/9 and varying by less than 1%), and almost flat (deviating from the planes zn by at most (Δz)/36). Away from the z-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. In the symmetry planes (e.g., y=0), the number of rings in the mth row, measured from the origin and before the transition to hairpins, is given by

36.7.7 nmax(m)=25613m26952.

Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane z=0 related by 2π/3 rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates (x=rcosθ,y=rsinθ) is given by

36.7.8 r=3((2n1)π4|sin(32θ)|)2/3(1+O(n1)),

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

The zeros of these functions are curves in 𝐱=(x,y,z) space; see Nye (2007) for Φ3 and Nye (2006) for Φ(H).