§36.2 Catastrophes and Canonical Integrals

Referenced by:: §36.11, §36.15(iii)
Permalink:: http://dlmf.nist.gov/36.2

§36.2(i) Definitions
§36.2(ii) Special Cases
§36.2(iii) Symmetries
§36.2(iv) Addendum to 36.2(ii) Special Cases

§36.2(i) Definitions

Notes:: The convergence of the oscillatory integrals (36.2.4)–(36.2.11) can be confirmed by rotating the integration paths in the complex plane. For (36.2.6) see Berry et al. (1979). For (36.2.7) shift the $s$ variable in (36.2.5) (with (36.2.2)) to remove the quadratic term, integrate, and then deform the contour of the remaining $t$ integration. For (36.2.8) see Berry and Howls (1990). For (36.2.9) integrate (36.2.5) (with (36.2.3)) with respect to $t$ .
Keywords:: canonical integrals
Referenced by:: §36.2(iii)
Permalink:: http://dlmf.nist.gov/36.2.i

¶ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

Keywords:: cusp catastophe, cuspoids, fold catastrophe, swallowtail catastrophe

36.2.1 $\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)=t^{{K+2}}+\sum _{{m=1}}^{K}x_{m}t^{m}.$

Defines:: $\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe
Symbols:: $n$ : integer, $t$ : variable, $K$ : codimension and $x_{i}$ : real parameter
Referenced by:: §36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E1
Encodings:: TeX, pMML, png

Special cases: $K=1$ , fold catastrophe; $K=2$ , cusp catastrophe; $K=3$ , swallowtail catastrophe.

¶ Normal Forms for Umbilic Catastrophes with Codimension $K=3$

Keywords:: elliptic umbilic catastrophe, hyperbolic umbilic catastrophe, umbilics

36.2.2 $\mathop{\Phi^{{(\mathrm{E})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt+xs,$ $\mathbf{x}=\{ x,y,z\}$ ,

Defines:: $\mathop{\Phi^{{(\mathrm{E})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$ : elliptic umbilic catastrophe
Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $y$ : real parameter, $z$ : real parameter, $t$ : variable, $s$ : variable and $x$ : real parameter
Referenced by:: §36.10(iii), §36.2(i), §36.7(iii), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E2
Encodings:: TeX, pMML, png

(elliptic umbilic).

36.2.3 $\mathop{\Phi^{{(\mathrm{H})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)=s^{3}+t^{3}+zst+yt+xs,$ $\mathbf{x}=\{ x,y,z\}$ ,

Defines:: $\mathop{\Phi^{{(\mathrm{H})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$ : hyperbolic umbilic catastrophe
Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $y$ : real parameter, $z$ : real parameter, $t$ : variable, $s$ : variable and $x$ : real parameter
Referenced by:: §36.10(iii), §36.2(i), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E3
Encodings:: TeX, pMML, png

(hyperbolic umbilic).

¶ Canonical Integrals

Keywords:: canonical integrals, cusp canonical integral, elliptic umbilic canonical integral, fold canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral

36.2.4 $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)=\int _{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(i\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)\right)dt.$

Defines:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral
Symbols:: $\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $t$ : variable and $K$ : codimension
Referenced by:: §36.12(i), §36.2(i), §36.7(ii), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E4
Encodings:: TeX, pMML, png

36.2.5 $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)=\int _{{-\infty}}^{\infty}\int _{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(i\mathop{\Phi^{{(\mathrm{U})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)\right)dsdt,$ $\mathrm{U}=\mathrm{E},\mathrm{H}$ .

Defines:: $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical umbilic integral
Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $t$ : variable and $s$ : variable
Referenced by:: §36.10(iii), §36.10(iv), §36.2(i), §36.7(iii), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E5
Encodings:: TeX, pMML, png

36.2.6 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)=2\sqrt{\ifrac{\pi}{3}}\,\mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{4}\pi\right)\right)\int _{{\infty\mathop{\exp\/}\nolimits\!\left(-7\pi i/12\right)}}^{{\infty\mathop{\exp\/}\nolimits\!\left(\pi i/12\right)}}\mathop{\exp\/}\nolimits\!\left(i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u^{2}}\right)\right)du,$

Defines:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral
Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Referenced by:: §36.2(i), §36.2(ii), §36.5(iii)
Permalink:: http://dlmf.nist.gov/36.2.E6
Encodings:: TeX, pMML, png

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

36.2.7

$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)=\dfrac{4\pi}{3^{{1/3}}}\mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{2}{27}z^{3}-\tfrac{1}{3}xz\right)\right)\left(\mathop{\exp\/}\nolimits\!\left(-i\dfrac{\pi}{6}\right)\mathrm{F}_{{+}}(\mathbf{x})+\mathop{\exp\/}\nolimits\!\left(i\dfrac{\pi}{6}\right)\mathrm{F}_{{-}}(\mathbf{x})\right),$

$\mathrm{F}_{{\pm}}(\mathbf{x})=\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(ry\mathop{\exp\/}\nolimits\!\left(\pm i\dfrac{\pi}{6}\right)\right)\mathop{\exp\/}\nolimits\!\left(2ir^{2}z\mathop{\exp\/}\nolimits\!\left(\pm i\dfrac{\pi}{3}\right)\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(3^{{2/3}}r^{2}+3^{{-1/3}}\mathop{\exp\/}\nolimits\!\left(\mp i\dfrac{\pi}{3}\right)\left(\tfrac{1}{3}z^{2}-x\right)\right)dr.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Referenced by:: §36.2(i)
Permalink:: http://dlmf.nist.gov/36.2.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

36.2.8 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)=4\sqrt{\ifrac{\pi}{6}}\,\mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*\int _{{\infty\mathop{\exp\/}\nolimits\!\left(5\pi i/12\right)}}^{{\infty\mathop{\exp\/}\nolimits\!\left(\pi i/12\right)}}\mathop{\exp\/}\nolimits\!\left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+x+y\right)u^{2}-\frac{(y-x)^{2}}{24u^{2}}\right)\right)du,$

Defines:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral
Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Referenced by:: §36.2(i), §36.2(ii), §36.5(iii)
Permalink:: http://dlmf.nist.gov/36.2.E8
Encodings:: TeX, pMML, png

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

36.2.9 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)=\frac{2\pi}{3^{{1/3}}}\int _{{\infty\mathop{\exp\/}\nolimits\!\left(5\pi i/6\right)}}^{{\infty\mathop{\exp\/}\nolimits\!\left(\pi i/6\right)}}\mathop{\exp\/}\nolimits\!\left(i(s^{3}+xs)\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{zs+y}{3^{{1/3}}}\right)ds.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $y$ : real parameter, $z$ : real parameter, $s$ : variable and $x$ : real parameter
Referenced by:: §36.2(i), §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E9
Encodings:: TeX, pMML, png

¶ Diffraction Catastrophes

Keywords:: diffraction catastrophes

36.2.10 $\mathop{\Psi _{{K}}\/}\nolimits\!(\mathbf{x};k)=\sqrt{k}\int _{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(ik\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)\right)dt,$ $k>0$ .

Defines:: $\mathop{\Psi _{{K}}\/}\nolimits\!(\mathbf{x};k)$ : diffraction catastrophe
Symbols:: $\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $k$ : variable, $t$ : variable and $K$ : codimension
Referenced by:: §36.12(i)
Permalink:: http://dlmf.nist.gov/36.2.E10
Encodings:: TeX, pMML, png

36.2.11 $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!(\mathbf{x};k)=k\int _{{-\infty}}^{\infty}\int _{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(ik\mathop{\Phi^{{(\mathrm{U})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)\right)dsdt,$ $\mathrm{U=E,H}$ ; $k>0$ .

Defines:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!(\mathbf{x};k)$ : elliptic umbilic canonical integral function, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!(\mathbf{x};k)$ : hyperbolic umbilic canonical integral function and $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!(\mathbf{x};k)$ : umbilic canonical integral function
Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $k$ : variable, $t$ : variable and $s$ : variable
Referenced by:: §36.2(i)
Permalink:: http://dlmf.nist.gov/36.2.E11
Encodings:: TeX, pMML, png

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

Notes:: For (36.2.12) and (36.2.13) use (4.10.11) and (9.5.4), respectively. For (36.2.15) and (36.2.17) use (5.9.1). For (36.2.18) combine (36.2.6), (36.2.8), and (5.9.1) For (36.2.19) use (12.5.1) and (12.14.13). For (36.2.20) see Trinkaus and Drepper (1977). For (36.2.21) use (36.2.9).
Keywords:: Airy functions, Pearcey integral, canonical integrals, fold canonical integral
Referenced by:: §36.2(iv)
Permalink:: http://dlmf.nist.gov/36.2.ii

36.2.12 $\mathop{\Psi _{{0}}\/}\nolimits=\sqrt{\pi}\mathop{\exp\/}\nolimits\!\left(i\frac{\pi}{4}\right).$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral and $\mathop{\exp\/}\nolimits z$ : exponential function
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E12
Encodings:: TeX, pMML, png

$\mathop{\Psi _{{1}}\/}\nolimits$ is related to the Airy function (§9.2):

36.2.13 $\mathop{\Psi _{{1}}\/}\nolimits\!\left(x\right)=\frac{2\pi}{3^{{1/3}}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{x}{3^{{1/3}}}\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral and $x$ : real parameter
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E13
Encodings:: TeX, pMML, png

$\mathop{\Psi _{{2}}\/}\nolimits$ is the Pearcey integral (Pearcey (1946)):

36.2.14 $\mathop{\Psi _{{2}}\/}\nolimits\!\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int _{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(i(t^{4}+x_{2}t^{2}+x_{1}t)\right)dt.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $t$ : variable and $x_{i}$ : real parameter
Referenced by:: §36.12(i)
Permalink:: http://dlmf.nist.gov/36.2.E14
Encodings:: TeX, pMML, png

(Other notations also appear in the literature.)

36.2.15 $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{K+2}\right)\*\begin{cases}\mathop{\exp\/}\nolimits\!\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \mathop{\cos\/}\nolimits\!\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function and $K$ : codimension
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E15
Encodings:: TeX, pMML, png

36.2.16

$\mathop{\Psi _{{1}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=1.54669,$

$\mathop{\Psi _{{2}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=1.67481+i\, 0.69373$

$\mathop{\Psi _{{3}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=1.74646,$

$\mathop{\Psi _{{4}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=1.79222+i\, 0.48022.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral
Permalink:: http://dlmf.nist.gov/36.2.E16
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

36.2.17

$\frac{{\partial}^{p}}{{\partial x_{1}}^{p}}\mathop{\Psi _{{K}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\mathop{\Gamma\/}\nolimits\!\left(\frac{p+1}{K+2}\right)\mathop{\cos\/}\nolimits\!\left(\frac{\pi}{2}\left(\frac{p+1}{K+2}+p\right)\right),$ $K$ odd,

$\frac{{\partial}^{2q+1}}{{\partial x_{1}}^{2q+1}}\mathop{\Psi _{{K}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=0,$ $K$ even,

$\frac{{\partial}^{2q}}{{\partial x_{1}}^{2q}}\mathop{\Psi _{{K}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\mathop{\Gamma\/}\nolimits\!\left(\frac{2q+1}{K+2}\right)\mathop{\exp\/}\nolimits\!\left(i\frac{\pi}{2}\left(\frac{2q+1}{K+2}+2q\right)\right),$ $K$ even.

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $K$ : codimension and $x_{i}$ : real parameter
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E17
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

36.2.18

$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\tfrac{1}{3}\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{6}\right)=3.28868,$

$\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(0\right)=\tfrac{1}{3}{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\tfrac{1}{3}\right)=2.39224.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral and $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E18
Encodings:: TeX, TeX, pMML, pMML, png, png

36.2.19 $\mathop{\Psi _{{2}}\/}\nolimits\!\left(0,y\right)=\frac{\pi}{2}\sqrt{\frac{|y|}{2}}\mathop{\exp\/}\nolimits\!\left(-i\frac{y^{2}}{8}\right)\left(\mathop{\exp\/}\nolimits\!\left(i\frac{\pi}{8}\right)\mathop{J_{{-\ifrac{1}{4}}}\/}\nolimits\!\left(\frac{y^{2}}{8}\right)-\mathop{\mathrm{sign}\/}\nolimits\!\left(y\right)\mathop{\exp\/}\nolimits\!\left(-i\frac{\pi}{8}\right)\mathop{J_{{\ifrac{1}{4}}}\/}\nolimits\!\left(\frac{y^{2}}{8}\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of $x$ and $y$ : real parameter
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E19
Encodings:: TeX, pMML, png

For the Bessel function $\mathop{J\/}\nolimits$ see §10.2(ii).

36.2.20 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,0\right)=2\pi^{2}(\tfrac{2}{3})^{{2/3}}\realpart{\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{x+iy}{12^{{1/3}}}\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\frac{x-iy}{12^{{1/3}}}\right)\right)},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\realpart{}$ : real part, $y$ : real parameter and $x$ : real parameter
Referenced by:: §36.2(ii), §36.7(iii), §36.8
Permalink:: http://dlmf.nist.gov/36.2.E20
Encodings:: TeX, pMML, png

36.2.21 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(x,y,0\right)=\frac{4\pi^{2}}{3^{{2/3}}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{x}{3^{{1/3}}}\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{y}{3^{{1/3}}}\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $y$ : real parameter and $x$ : real parameter
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.E21
Encodings:: TeX, pMML, png

Addendum: For further special cases see §36.2(iv)

§36.2(iii) Symmetries

Notes:: These results follow from the definitions given in §36.2(i).
Keywords:: canonical integrals, symmetries
Permalink:: http://dlmf.nist.gov/36.2.iii

36.2.22 $\mathop{\Psi _{{2K}}\/}\nolimits\!\left(\mathbf{x}^{{\prime}}\right)=\mathop{\Psi _{{2K}}\/}\nolimits\!\left(\mathbf{x}\right),$ $x_{{2m+1}}^{{\prime}}=-x_{{2m+1}}$ , $x_{{2m}}^{{\prime}}=x_{{2m}}$ .

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $n$ : integer, $K$ : codimension and $x_{i}$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E22
Encodings:: TeX, pMML, png

36.2.23 $\mathop{\Psi _{{2K+1}}\/}\nolimits\!\left(\mathbf{x}^{{\prime}}\right)={\mathop{\Psi _{{2K+1}}\/}\nolimits^{{\ast}}}\!\left(\mathbf{x}\right),$ $x_{{2m+1}}^{{\prime}}=x_{{2m+1}}$ , $x_{{2m}}^{{\prime}}=-x_{{2m}}$ .

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $n$ : integer, $K$ : codimension and $x_{i}$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E23
Encodings:: TeX, pMML, png

36.2.24 $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(x,y,z\right)={\Psi^{{\ast}}}^{{(\mathrm{U})}}(x,y,-z),$ $\mathrm{U=E,H}$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical umbilic integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E24
Encodings:: TeX, pMML, png

36.2.25 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,-y,z\right)=\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,z\right).$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E25
Encodings:: TeX, pMML, png

36.2.26 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt{3}}{2}x-\tfrac{1}{2}y,z\right)=\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,z\right),$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E26
Encodings:: TeX, pMML, png

(rotation by $\pm\tfrac{2}{3}\pi$ in $x,y$ plane).

36.2.27 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(x,y,z\right)=\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(y,x,z\right).$

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.2.E27
Encodings:: TeX, pMML, png

§36.2(iv) Addendum to 36.2(ii) Special Cases

Notes:: See Berry and Howls (2010). (The material in this subsection was added in Version 1.0.5. It will be incorporated in the next print edition.)
Keywords:: canonical integrals
Referenced by:: §36.2(ii)
Permalink:: http://dlmf.nist.gov/36.2.iv

36.2.28 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(0,0,z\right)={\Psi^{{\ast}}}^{{(\mathrm{E})}}(0,0,-z)\\ =2\pi\sqrt{\frac{\pi z}{27}}\mathop{\exp\/}\nolimits\!\left(\frac{2}{27}iz^{3}\right)\*\left(\mathop{J_{{-1/6}}\/}\nolimits\!\left(\frac{2}{27}z^{3}\right)+i\mathop{J_{{1/6}}\/}\nolimits\!\left(\frac{2}{27}z^{3}\right)\right),$ $z\geq 0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function and $z$ : real parameter
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/36.2.E28
Encodings:: TeX, pMML, png
Addition (effective with 1.0.5):: This equation has been added. For the proof see Berry and Howls (2010).
Reported 2012-04-02

36.2.29 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(0,0,z\right)={\Psi^{{\ast}}}^{{(\mathrm{H})}}(0,0,-z)=\frac{2^{{1/3}}}{\sqrt{3}}\mathop{\exp\/}\nolimits\!\left(\frac{1}{27}iz^{3}\right)\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(0,0,-\frac{z}{2^{{2/3}}}\right),$ $-\infty<z<\infty$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function and $z$ : real parameter
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/36.2.E29
Encodings:: TeX, pMML, png
Addition (effective with 1.0.5):: This equation has been added. For the proof see Berry and Howls (2010).
Reported 2012-05-01

Here the functions ${\Psi^{{\ast}}}^{{(\mathrm{E})}}$ and ${\Psi^{{\ast}}}^{{(\mathrm{H})}}$ are the complex conjugates of the functions $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits$ and $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits$ , respectively.