36 Integrals with Coalescing SaddlesProperties36.6 Scaling Relations36.8 Convergent Series Expansions

- §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

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

The zeros in Table 36.7.1 are points in the $\mathbf{x}=\left(x,y\right)$ plane, where $\mathrm{ph}{\mathrm{\Psi}}_{2}\left(\mathbf{x}\right)$ is undetermined. All zeros have $$, and fall into two classes. Inside the cusp, that is, for $$, 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 | ${y}_{m}$ | $=-\sqrt{2\pi \left(2m+1\right)},$ | ||

$m=1,2,3,\mathrm{\dots}$, | ||||

${x}_{m,n}^{\pm}$ | $=\sqrt{{\displaystyle \frac{2}{-{y}_{m}}}}\left(2n+\frac{1}{2}+{\left(-1\right)}^{m}\frac{1}{2}\pm \frac{1}{4}\right)\pi ,$ | |||

$m=1,2,3,\mathrm{\dots}$, $n=0,\pm 1,\pm 2,\mathrm{\dots}$. | ||||

Zeros $\left\{{\scriptscriptstyle \begin{array}{c}x\\ y\end{array}}\right\}$ inside, and zeros $\left[{\scriptscriptstyle \begin{array}{c}x\\ y\end{array}}\right]$ outside, the cusp ${x}^{2}=\frac{8}{27}{\left|y\right|}^{3}$. | |||||
---|---|---|---|---|---|

$\left\{{\displaystyle \begin{array}{c}\pm 0.52768\\ -4.37804\end{array}}\right\}$ | $\left[{\displaystyle \begin{array}{c}\pm 2.35218\\ -1.74360\end{array}}\right]$ | ||||

$\left\{{\displaystyle \begin{array}{c}\pm 1.41101\\ -5.55470\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 2.36094\\ -5.52321\end{array}}\right\}$ | $\left[{\displaystyle \begin{array}{c}\pm 4.42707\\ -3.05791\end{array}}\right]$ | |||

$\left\{{\displaystyle \begin{array}{c}\pm 0.43039\\ -6.64285\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 3.06389\\ -6.44624\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 3.95806\\ -6.40312\end{array}}\right\}$ | $\left[{\displaystyle \begin{array}{c}\pm 6.16185\\ -4.03551\end{array}}\right]$ | ||

$\left\{{\displaystyle \begin{array}{c}\pm 1.21605\\ -7.49906\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 2.02922\\ -7.48629\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 4.56537\\ -7.19629\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 5.42206\\ -7.14718\end{array}}\right\}$ | $\left[{\displaystyle \begin{array}{c}\pm 7.72352\\ -4.84817\end{array}}\right]$ | |

$\left\{{\displaystyle \begin{array}{c}\pm 0.38488\\ -8.31916\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 2.71193\\ -8.22315\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 3.49286\\ -8.20326\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 5.96669\\ -7.85723\end{array}}\right\}$ | $\left\{{\displaystyle \begin{array}{c}\pm 6.79538\\ -7.80456\end{array}}\right\}$ | $\left[{\displaystyle \begin{array}{c}\pm 9.17308\\ -5.55831\end{array}}\right]$ |

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

36.7.2 | ${x}_{n}$ | $=\pm {\left({\displaystyle \frac{8}{27}}\right)}^{1/2}{\left|{y}_{n}\right|}^{3/2}\left(1+{\xi}_{n}\right),$ | ||

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

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

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

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

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

$n=1,2,3,\mathrm{\dots}$. | |||

Near $z={z}_{n}$, and for small $x$ and $y$, the modulus $\left|{\mathrm{\Psi}}^{\left(\mathrm{E}\right)}\left(\mathbf{x}\right)\right|$ 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 | $\Delta z$ | $={\displaystyle \frac{9\pi}{2{z}_{n}^{2}}},$ | ||

$\Delta x$ | $={\displaystyle \frac{6\pi}{{z}_{n}}}.$ | |||

The zeros are approximated by solutions of the equation

36.7.6 | $$\mathrm{exp}\left(-2\pi \mathrm{i}\left(\frac{z-{z}_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)\right)\left(2\mathrm{exp}\left(\frac{-6\pi \mathrm{i}x}{\Delta x}\right)\mathrm{cos}\left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+1\right)=\sqrt{3}.$$ | ||

The rings are almost circular (radii close to $\left(\Delta x\right)/9$ and varying by less than 1%), and almost flat (deviating from the planes ${z}_{n}$ by at most $\left(\Delta z\right)/36$). 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 | $${n}_{max}\left(m\right)=\lfloor \frac{256}{13}m-\frac{269}{52}\rfloor .$$ | ||

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 $\left(x=r\mathrm{cos}\theta ,\hspace{0.25em}y=r\mathrm{sin}\theta \right)$ is given by

36.7.8 | $$r=3{\left(\frac{\left(2n-1\right)\pi}{4\left|\mathrm{sin}\left(\frac{3}{2}\theta \right)\right|}\right)}^{2/3}\left(1+O\left({n}^{-1}\right)\right),$$ | ||

$n\to \mathrm{\infty}$. | |||