36 Integrals with Coalescing SaddlesProperties36.9 Integral Identities36.11 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

Contents

1. §36.10(i) Equations for ${\mathrm{\Psi }}_K\left(𝐱\right)$
2. §36.10(ii) Partial Derivatives with Respect to the $x_n$
3. §36.10(iii) Operator Equations
4. §36.10(iv) Partial $z$-Derivatives

§36.10(i) Equations for ${\mathrm{\Psi }}_K\left(𝐱\right)$

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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 ${\mathrm{\Psi }}_K\left(𝐱\right)$ satisfy the operator equation

36.10.1		$${\mathrm{\Phi }}_K^{\prime }(-\mathrm{i}\frac{\partial }{\partial x_1};𝐱){\mathrm{\Psi }}_K\left(𝐱\right)=0,$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $K$: codimension and $x_{\mathrm{i}}$: 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		$$\frac{{\partial }^{K+1}{\mathrm{\Psi }}_K\left(𝐱\right)}{{\partial x_1}^{K+1}}+\sum _{m=1}^K{(-\mathrm{i})}^{m-K-2}\left(\frac{m{x}_m}{K+2}\right)\frac{{\partial }^{m-1}{\mathrm{\Psi }}_K\left(𝐱\right)}{{\partial x_1}^{m-1}}=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $n$: integer, $K$: codimension and $x_{\mathrm{i}}$: 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		$$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$\frac{{\partial }^3{\mathrm{\Psi }}_2}{{\partial x}^3}-\frac{1}{2}y\frac{\partial {\mathrm{\Psi }}_2}{\partial x}-\frac{\mathrm{i}}{4}x{\mathrm{\Psi }}_2=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$\frac{{\partial }^4{\mathrm{\Psi }}_3}{{\partial x}^4}-\frac{3}{5}z\frac{{\partial }^2{\mathrm{\Psi }}_3}{{\partial x}^2}-\frac{2\mathrm{i}}{5}y\frac{\partial {\mathrm{\Psi }}_3}{\partial x}+\frac{1}{5}x{\mathrm{\Psi }}_3=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 $x_n$

ⓘ

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		$$\frac{{\partial }^{ln}{\mathrm{\Psi }}_K}{{\partial x_m}^{ln}}={\mathrm{i}}^{n(l-m)}\frac{{\partial }^{mn}{\mathrm{\Psi }}_K}{{\partial x_l}^{mn}},$$
$1\le m\le K$, $1\le l\le K$.
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $l$: integer, $n$: integer, $m$: integer, $K$: codimension and $x_{\mathrm{i}}$: 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		$$\frac{{\partial }^{2n}{\mathrm{\Psi }}_2}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_2}{{\partial y}^n}.$$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		${\displaystyle \frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial x}^{2n}}}$	$={\mathrm{i}}^n{\displaystyle \frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial y}^n}},$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		${\displaystyle \frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial x}^{3n}}}$	$={(-1)}^n{\displaystyle \frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial z}^n}},$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		${\displaystyle \frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial y}^{3n}}}$	$={\mathrm{i}}^n{\displaystyle \frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial z}^{2n}}}.$
ⓘ Symbols: ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations

36.10.11		${\mathrm{\Phi }}_s^{(\mathrm{U})}(-\mathrm{i}{\displaystyle \frac{\partial }{\partial x}},-\mathrm{i}{\displaystyle \frac{\partial }{\partial y}};𝐱){\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$	$=0,$
		${\mathrm{\Phi }}_t^{(\mathrm{U})}(-\mathrm{i}{\displaystyle \frac{\partial }{\partial x}},-\mathrm{i}{\displaystyle \frac{\partial }{\partial y}};𝐱){\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$	$=0,$
ⓘ Symbols: ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$: 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		${\mathrm{\Phi }}_s^{(\mathrm{U})}(s,t;𝐱)$
		${\mathrm{\Phi }}_t^{(\mathrm{U})}(s,t;𝐱)$
ⓘ Symbols: ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$, $t$: 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		$$6\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x\partial y}-2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial y}+y{\mathrm{\Psi }}^{(\mathrm{E})}=0,$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$: $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0.$$ See also: Annotations for §36.10(iii), §36.10 and Ch.36

36.10.15		$$3\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{{\partial x}^2}+\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial y}-x{\mathrm{\Psi }}^{(\mathrm{H})}=0,$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$3\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{{\partial y}^2}+\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}-y{\mathrm{\Psi }}^{(\mathrm{H})}=0.$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$\mathrm{i}\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial z}=\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}+\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2},$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$\mathrm{i}\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial z}=\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x\partial y}.$$
ⓘ Symbols: ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 Identities36.11 Leading-Order Asymptotics

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