# §36.7(i) Fold Canonical Integral

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

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

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 $\displaystyle x_{n}$ $\displaystyle=\pm\left(\dfrac{8}{27}\right)^{1/2}|y_{n}|^{3/2}(1+\xi_{n}),$ $\displaystyle y_{n}$ $\displaystyle=-\left(\frac{3\pi(8n+5)}{9+8\xi_{n}}\right)^{1/2},$ 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

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 $\displaystyle\Delta z$ $\displaystyle=\frac{9\pi}{2z_{n}^{2}},$ $\displaystyle\Delta x$ $\displaystyle=\frac{6\pi}{z_{n}}.$ 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 $\mathop{\exp\/}\nolimits\!\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x% }{\Delta x}\right)\right)\*{\left(2\mathop{\exp\/}\nolimits\!\left(\frac{-6\pi ix% }{\Delta x}\right)\mathop{\cos\/}\nolimits\!\left(\frac{2\sqrt{3}\pi y}{\Delta x% }\right)+1\right)}=\sqrt{3}.$

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

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

# §36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

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 $\mathop{\Phi^{(\mathrm{H})}\/}\nolimits$.