§36.7 Zeros

Keywords:: canonical integrals
Referenced by:: §36.15(ii)
Permalink:: http://dlmf.nist.gov/36.7

§36.7(i) Fold Canonical Integral
§36.7(ii) Cusp Canonical Integral
§36.7(iii) Elliptic Umbilic Canonical Integral
§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

§36.7(i) Fold Canonical Integral

Keywords:: fold canonical integral, fold catastrophe
Permalink:: http://dlmf.nist.gov/36.7.i

This is the Airy function $\mathop{\mathrm{Ai}\/}\nolimits$ (§9.2).

§36.7(ii) Cusp Canonical Integral

Notes:: See Berry et al. (1979). (36.7.2) and (36.7.3) may be derived by setting to zero the stationary-phase approximation (§2.3(iv)) of the Pearcey integral $\mathop{\Psi _{{2}}\/}\nolimits\!\left(x,y\right)$ just outside the caustic; this involves one real saddle and one complex saddle. Table 36.7.1 was computed by the authors.
Keywords:: Pearcey integral, cusp canonical integral
Permalink:: http://dlmf.nist.gov/36.7.ii

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

The zeros in Table 36.7.1 are points in the $\mathbf{x}=(x,y)$ plane, where $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi _{{2}}\/}\nolimits\!\left(\mathbf{x}\right)$ is undetermined. All zeros have $y<0$ , and fall into two classes. Inside the cusp, that is, for $x^{2}<8|y|^{3}/27$ , 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

$y_{m}=-\sqrt{2\pi(2m+1)},$ $m=1,2,3,\dots$ ,

$x_{{m,n}}^{{\pm}}=\sqrt{\dfrac{2}{-y_{m}}}\left(2n+\tfrac{1}{2}+(-1)^{m}\tfrac{1}{2}\pm\tfrac{1}{4}\right)\pi,$ $m=1,2,3,\dots$ , $n=0,\pm 1,\pm 2,\dots$ .

Symbols:: $y$ : real parameter, $n$ : integer, $m$ : integer and $x_{i}$ : real parameter
Referenced by:: §36.7(iii)
Permalink:: http://dlmf.nist.gov/36.7.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D.

Zeros $\genfrac{\{}{\}}{0.0pt}{}{x}{y}$ inside, and zeros $\genfrac{[}{]}{0.0pt}{}{x}{y}$ outside, the cusp $x^{2}=\tfrac{8}{27}\|y\|^{3}$ .

$\genfrac{\{}{\}}{0.0pt}{0}{\pm 0.52768}{-4.37804}$	$\genfrac{[}{]}{0.0pt}{0}{\pm 2.35218}{-1.74360}$
$\genfrac{\{}{\}}{0.0pt}{0}{\pm 1.41101}{-5.55470}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 2.36094}{-5.52321}$	$\genfrac{[}{]}{0.0pt}{0}{\pm 4.42707}{-3.05791}$
$\genfrac{\{}{\}}{0.0pt}{0}{\pm 0.43039}{-6.64285}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 3.06389}{-6.44624}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 3.95806}{-6.40312}$	$\genfrac{[}{]}{0.0pt}{0}{\pm 6.16185}{-4.03551}$
$\genfrac{\{}{\}}{0.0pt}{0}{\pm 1.21605}{-7.49906}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 2.02922}{-7.48629}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 4.56537}{-7.19629}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 5.42206}{-7.14718}$	$\genfrac{[}{]}{0.0pt}{0}{\pm 7.72352}{-4.84817}$
$\genfrac{\{}{\}}{0.0pt}{0}{\pm 0.38488}{-8.31916}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 2.71193}{-8.22315}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 3.49286}{-8.20326}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 5.96669}{-7.85723}$	$\genfrac{\{}{\}}{0.0pt}{0}{\pm 6.79538}{-7.80456}$	$\genfrac{[}{]}{0.0pt}{0}{\pm 9.17308}{-5.55831}$

Symbols:: $y$ : real parameter and $x$ : real parameter
Keywords:: Pearcey integral, cusp canonical integral
Referenced by:: §36.7(ii), §36.7(ii)
Permalink:: http://dlmf.nist.gov/36.7.T1

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

36.7.2

$x_{n}=\pm\left(\dfrac{8}{27}\right)^{{1/2}}|y_{n}|^{{3/2}}(1+\xi _{n}),$

$y_{n}=-\left(\frac{3\pi(8n+5)}{9+8\xi _{n}}\right)^{{1/2}},$

Symbols:: $y$ : real parameter, $m$ : integer and $x_{i}$ : real parameter
Referenced by:: §36.7(ii), §36.7(iii)
Permalink:: http://dlmf.nist.gov/36.7.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

where $\xi _{n}$ is the real solution of

36.7.3 $\frac{3\pi(8n+5)}{9+8\xi _{n}}\xi _{n}^{{3/2}}=\dfrac{27}{16}\left(\dfrac{3}{2}\right)^{{1/2}}\left(\mathop{\ln\/}\nolimits\!\left(\frac{1}{\xi _{n}}\right)+3\mathop{\ln\/}\nolimits\!\left(\dfrac{3}{2}\right)\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $m$ : integer
Referenced by:: §36.7(ii)
Permalink:: http://dlmf.nist.gov/36.7.E3
Encodings:: TeX, pMML, png

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

§36.7(iii) Elliptic Umbilic Canonical Integral

Notes:: See Berry et al. (1979).
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.7.iii

This is (36.2.5) with (36.2.2). The zeros are lines in $\mathbf{x}=(x,y,z)$ space where $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ 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 $z_{n}=\pm 3(\tfrac{1}{4}\pi(2n-\tfrac{1}{2}))^{{1/3}}\\ =3.48734(n-\tfrac{1}{4})^{{1/3}},$ $n=1,2,3,\dots$ .

Symbols:: $z$ : real parameter and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E4
Encodings:: TeX, pMML, png

Near $z=z_{n}$ , and for small $x$ and $y$ , the modulus $|\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)|$ 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

$\Delta z=\frac{9\pi}{2z_{n}^{2}},$

$\Delta x=\frac{6\pi}{z_{n}}.$

Symbols:: $z$ : real parameter, $m$ : integer and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.7.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

The zeros are approximated by solutions of the equation

36.7.6 $\mathop{\exp\/}\nolimits\!\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)\right)\*{\left(2\mathop{\exp\/}\nolimits\!\left(\frac{-6\pi ix}{\Delta x}\right)\mathop{\cos\/}\nolimits\!\left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+1\right)}=\sqrt{3}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $y$ : real parameter, $z$ : real parameter, $m$ : integer and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.7.E6
Encodings:: TeX, pMML, png

The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_{n}$ by at most $(\Delta z)/36$ ). 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 $m$ th row, measured from the origin and before the transition to hairpins, is given by

36.7.7 $n_{{\max}}(m)=\left\lfloor\tfrac{256}{13}m-\tfrac{269}{52}\right\rfloor.$

Symbols:: $\left\lfloor x\right\rfloor$ : floor of $x$ , $n$ : integer and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E7
Encodings:: TeX, pMML, png

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 $2\pi/3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\mathop{\cos\/}\nolimits\theta,\; y=r\mathop{\sin\/}\nolimits\theta)$ is given by

36.7.8 $r=3\left(\frac{(2n-1)\pi}{4|\mathop{\sin\/}\nolimits\!\left(\tfrac{3}{2}\theta\right)|}\right)^{{2/3}}(1+\mathop{O\/}\nolimits\!\left(n^{{-1}}\right)),$ $n\to\infty$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\sin\/}\nolimits z$ : sine function and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E8
Encodings:: TeX, pMML, png

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

Keywords:: canonical integrals, hyperbolic umbilic canonical integral, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.7.iv

The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\mathop{\Phi _{{3}}\/}\nolimits$ and Nye (2006) for $\mathop{\Phi^{{(\mathrm{H})}}\/}\nolimits$ .

§36.7 Zeros

Contents

§36.7(i) Fold Canonical Integral

§36.7(ii) Cusp Canonical Integral

§36.7(iii) Elliptic Umbilic Canonical Integral

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals