36 Integrals with Coalescing Saddles Properties 36.9 Integral Identities 36.11 Leading-Order Asymptotics

§36.10 Differential Equations

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Keywords:: canonical integrals, differential equations
Permalink:: http://dlmf.nist.gov/36.10
See also:: Annotations for Ch.36

§36.10(i) Equations for $\Psi_{K}\left(\mathbf{x}\right)$

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Notes:: See Connor et al. (1983).
Permalink:: http://dlmf.nist.gov/36.10.i
See also:: Annotations for §36.10 and Ch.36

In terms of the normal form (36.2.1) the $\Psi_{K}\left(\mathbf{x}\right)$ satisfy the operator equation

36.10.1		$\Phi_{K}'\left(-i\frac{\partial}{\partial x_{1}};\mathbf{x}\right)\Psi_{K}% \left(\mathbf{x}\right)=0,$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $K$ : codimension and $x_{i}$ : real parameter Referenced by: §36.10(i) Permalink: http://dlmf.nist.gov/36.10.E1 Encodings: TeX, pMML, png See also: Annotations for §36.10(i), §36.10 and Ch.36

or explicitly,

36.10.2		$\frac{{\partial}^{K+1}\Psi_{K}\left(\mathbf{x}\right)}{{\partial x_{1}}^{K+1}}% +\sum_{m=1}^{K}(-i)^{m-K-2}\left(\frac{mx_{m}}{K+2}\right)\frac{{\partial}^{m-% 1}\Psi_{K}\left(\mathbf{x}\right)}{{\partial x_{1}}^{m-1}}=0.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : integer, $K$ : codimension and $x_{i}$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E2 Encodings: TeX, pMML, png See also: Annotations for §36.10(i), §36.10 and Ch.36

Special Cases

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Keywords:: differential equation, fold canonical integral
See also:: Annotations for §36.10(i), §36.10 and Ch.36

$K=1$ , fold: (36.10.1) becomes Airy’s equation (§9.2(i))

36.10.3		$\frac{{\partial}^{2}\Psi_{1}}{{\partial x}^{2}}-\frac{x}{3}\Psi_{1}=0.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E3 Encodings: TeX, pMML, png See also: Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

$K=2$ , cusp:

36.10.4		$\frac{{\partial}^{3}\Psi_{2}}{{\partial x}^{3}}-\frac{1}{2}y\frac{\partial\Psi% _{2}}{\partial x}-\frac{i}{4}x\Psi_{2}=0.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E4 Encodings: TeX, pMML, png See also: Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

$K=3$ , swallowtail:

36.10.5		$\frac{{\partial}^{4}\Psi_{3}}{{\partial x}^{4}}-\frac{3}{5}z\frac{{\partial}^{% 2}\Psi_{3}}{{\partial x}^{2}}-\frac{2i}{5}y\frac{\partial\Psi_{3}}{\partial x}% +\frac{1}{5}x\Psi_{3}=0.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E5 Encodings: TeX, pMML, png See also: Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

§36.10(ii) Partial Derivatives with Respect to the $x_{n}$

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Notes:: See Connor et al. (1983).
Permalink:: http://dlmf.nist.gov/36.10.ii
See also:: Annotations for §36.10 and Ch.36

36.10.6		$\frac{{\partial}^{ln}\Psi_{K}}{{\partial x_{m}}^{ln}}=i^{n(l-m)}\frac{{% \partial}^{mn}\Psi_{K}}{{\partial x_{l}}^{mn}},$
$1\leq m\leq K$ , $1\leq l\leq K$ .
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $l$ : integer, $n$ : integer, $m$ : integer, $K$ : codimension and $x_{i}$ : real parameter Referenced by: §36.10(ii) Permalink: http://dlmf.nist.gov/36.10.E6 Encodings: TeX, pMML, png See also: Annotations for §36.10(ii), §36.10 and Ch.36

Special Cases

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Keywords:: Pearcey integral, differential equation, differential equations, swallowtail canonical integral
See also:: Annotations for §36.10(ii), §36.10 and Ch.36

$K=1$ , fold: (36.10.6) is an identity.

$K=2$ , cusp:

36.10.7		$\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $m$ : integer and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E7 Encodings: TeX, pMML, png See also: Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

$K=3$ , swallowtail:

36.10.8	$\displaystyle\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}$	$\displaystyle=i^{n}\frac{{\partial}^{n}\Psi_{3}}{{\partial y}^{n}},$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $m$ : integer and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E8 Encodings: TeX, pMML, png See also: Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36
36.10.9	$\displaystyle\frac{{\partial}^{3n}\Psi_{3}}{{\partial x}^{3n}}$	$\displaystyle=(-1)^{n}\frac{{\partial}^{n}\Psi_{3}}{{\partial z}^{n}},$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : real parameter, $m$ : integer and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E9 Encodings: TeX, pMML, png See also: Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36
36.10.10	$\displaystyle\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}$	$\displaystyle=i^{n}\frac{{\partial}^{2n}\Psi_{3}}{{\partial z}^{2n}}.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $m$ : integer Permalink: http://dlmf.nist.gov/36.10.E10 Encodings: TeX, pMML, png See also: Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

§36.10(iii) Operator Equations

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Keywords:: differential equations, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral
Notes:: Use repeated differentiations with respect to $x$ or $y$ , in combinations that generate exact derivatives of the exponents in (36.2.5).
Permalink:: http://dlmf.nist.gov/36.10.iii
See also:: Annotations for §36.10 and Ch.36

In terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations

36.10.11	$\displaystyle{\Phi_{s}}^{(\mathrm{U})}\left(-i\frac{\partial}{\partial x},-i% \frac{\partial}{\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{% x}\right)$	$\displaystyle=0,$
36.10.11	$\displaystyle{\Phi_{t}}^{(\mathrm{U})}\left(-i\frac{\partial}{\partial x},-i% \frac{\partial}{\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{% x}\right)$	$\displaystyle=0,$
ⓘ Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)$ : umbilic canonical integral function, $y$ : real parameter, $t$ : variable, $s$ : variable and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.10(iii), §36.10 and Ch.36

where

36.10.12	$\displaystyle{\Phi_{s}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)$	$\displaystyle=\frac{\partial}{\partial s}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{% x}\right),$
36.10.12	$\displaystyle{\Phi_{t}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)$	$\displaystyle=\frac{\partial}{\partial t}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{% x}\right).$
ⓘ Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $t$ : variable and $s$ : variable Permalink: http://dlmf.nist.gov/36.10.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.10(iii), §36.10 and Ch.36

Explicitly,

36.10.13		$6\frac{\,{\partial}^{2}\Psi^{(\mathrm{E})}}{\,\partial x\,\partial y}-2iz\frac% {\partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
ⓘ Symbols: $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E13 Encodings: TeX, pMML, png See also: Annotations for §36.10(iii), §36.10 and Ch.36

36.10.14

3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2iz\frac{\partial% \Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0.

ⓘ

Symbols:

\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)

: elliptic umbilic canonical integral function,

\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)

: hyperbolic umbilic canonical integral function,

\mathrm{i}

: imaginary unit,

\frac{\partial\NVar{f}}{\partial\NVar{x}}

: partial derivative of

f

with respect to

x

\,\partial\NVar{x}

: partial differential of

x

y

: real parameter,

z

: real parameter and

x

: real parameter

Referenced by:

Erratum (V1.0.1) for Equation (36.10.14)

Permalink:

http://dlmf.nist.gov/36.10.E14

Encodings:

TeX, pMML, png

Errata (effective with 1.0.1):

Originally appeared with

\frac{\partial\Psi^{(\mathrm{H})}}{\partial x}

, rather than

\frac{\partial\Psi^{(\mathrm{E})}}{\partial x}

3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-x\Psi^{(\mathrm{E})}=0.

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

36.10.15		$3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
ⓘ Symbols: $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E15 Encodings: TeX, pMML, png See also: Annotations for §36.10(iii), §36.10 and Ch.36

36.10.16		$3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$
ⓘ Symbols: $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E16 Encodings: TeX, pMML, png See also: Annotations for §36.10(iii), §36.10 and Ch.36

§36.10(iv) Partial $z$ -Derivatives

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Keywords:: paraxial wave equation
Notes:: Use repeated differentiations with respect to $x, y$ , or $z$ , in combinations that generate exact derivatives of the exponents in (36.2.5).
Permalink:: http://dlmf.nist.gov/36.10.iv
See also:: Annotations for §36.10 and Ch.36

36.10.17		$i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
ⓘ Symbols: $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.10(iv) Permalink: http://dlmf.nist.gov/36.10.E17 Encodings: TeX, pMML, png See also: Annotations for §36.10(iv), §36.10 and Ch.36

36.10.18		$i\frac{\partial\Psi^{(\mathrm{H})}}{\partial z}=\frac{\,{\partial}^{2}\Psi^{(% \mathrm{H})}}{\,\partial x\,\partial y}.$
ⓘ Symbols: $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.10.E18 Encodings: TeX, pMML, png See also: Annotations for §36.10(iv), §36.10 and Ch.36

Equation (36.10.17) is the paraxial wave equation.

36.9 Integral Identities 36.11 Leading-Order Asymptotics

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§36.10 Differential Equations

Contents

§36.10(i) Equations for ΨK⁡(𝐱)

Special Cases

§36.10(ii) Partial Derivatives with Respect to the xn

Special Cases

§36.10(iii) Operator Equations

§36.10(iv) Partial z-Derivatives

§36.10(i) Equations for $\Psi_{K}\left(\mathbf{x}\right)$

§36.10(ii) Partial Derivatives with Respect to the $x_{n}$

§36.10(iv) Partial $z$ -Derivatives