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

§36.10 Differential Equations

Keywords:: canonical integrals, differential equations
Permalink:: http://dlmf.nist.gov/36.10

§36.10(i) Equations for $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$
§36.10(ii) Partial Derivatives with Respect to the $x_{n}$
§36.10(iii) Operator Equations
§36.10(iv) Partial -Derivatives

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

Notes:: See Connor et al. (1983).
Permalink:: http://dlmf.nist.gov/36.10.i

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

36.10.1 ${\mathop{\Phi_{{K}}\/}\nolimits^{{\prime}}}\!\left(-i\frac{\partial}{\partial x% _{1}};\mathbf{x}\right)\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)% =0,$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : codimension and $x_{i}$ : real parameter
Referenced by:: ¶ ‣ §36.10(i)
Permalink:: http://dlmf.nist.gov/36.10.E1
Encodings:: TeX, pMML, png

or explicitly,

36.10.2 $\frac{{\partial}^{K+1}\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)}% {{\partial x_{1}}^{K+1}}+\sum_{{m=1}}^{K}(-i)^{{m-K-2}}\left(\frac{mx_{m}}{K+2% }\right)\frac{{\partial}^{m-1}\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}% \right)}{{\partial x_{1}}^{m-1}}=0.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : codimension and $x_{i}$ : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E2
Encodings:: TeX, pMML, png

¶ Special Cases

Keywords:: fold canonical integral

, fold: (36.10.1) becomes Airy’s equation (§9.2(i))

36.10.3 $\frac{{\partial}^{2}\mathop{\Psi_{{1}}\/}\nolimits}{{\partial x}^{2}}-\frac{x}% {3}\mathop{\Psi_{{1}}\/}\nolimits=0.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E3
Encodings:: TeX, pMML, png

, cusp:

36.10.4 $\frac{{\partial}^{3}\mathop{\Psi_{{2}}\/}\nolimits}{{\partial x}^{3}}-\frac{1}% {2}y\frac{\partial\mathop{\Psi_{{2}}\/}\nolimits}{\partial x}-\frac{i}{4}x% \mathop{\Psi_{{2}}\/}\nolimits=0.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E4
Encodings:: TeX, pMML, png

, swallowtail:

36.10.5 $\frac{{\partial}^{4}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial x}^{4}}-\frac{3}% {5}z\frac{{\partial}^{2}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial x}^{2}}-% \frac{2i}{5}y\frac{\partial\mathop{\Psi_{{3}}\/}\nolimits}{\partial x}+\frac{1% }{5}x\mathop{\Psi_{{3}}\/}\nolimits=0.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E5
Encodings:: TeX, pMML, png

§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

36.10.6 $\frac{{\partial}^{ln}\mathop{\Psi_{{K}}\/}\nolimits}{{\partial x_{m}}^{ln}}=i^% {{n(l-m)}}\frac{{\partial}^{mn}\mathop{\Psi_{{K}}\/}\nolimits}{{\partial x_{l}% }^{mn}},$ $1\leq m\leq K$ , $1\leq l\leq K$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : integer, : integer, : codimension and $x_{i}$ : real parameter
Referenced by:: ¶ ‣ §36.10(ii)
Permalink:: http://dlmf.nist.gov/36.10.E6
Encodings:: TeX, pMML, png

¶ Special Cases

Keywords:: Pearcey integral, differential equations, swallowtail canonical integral

, fold: (36.10.6) is an identity.

, cusp:

36.10.7 $\frac{{\partial}^{2n}\mathop{\Psi_{{2}}\/}\nolimits}{{\partial x}^{2n}}=i^{n}% \frac{{\partial}^{n}\mathop{\Psi_{{2}}\/}\nolimits}{{\partial y}^{n}}.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : integer and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E7
Encodings:: TeX, pMML, png

, swallowtail:

36.10.8 $\frac{{\partial}^{2n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial x}^{2n}}=i^{n}% \frac{{\partial}^{n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial y}^{n}},$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : integer and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E8
Encodings:: TeX, pMML, png

36.10.9 $\frac{{\partial}^{3n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial x}^{3n}}=(-1)^{% n}\frac{{\partial}^{n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial z}^{n}},$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : integer and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E9
Encodings:: TeX, pMML, png

36.10.10 $\frac{{\partial}^{3n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial y}^{3n}}=i^{n}% \frac{{\partial}^{2n}\mathop{\Psi_{{3}}\/}\nolimits}{{\partial z}^{2n}}.$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : integer
Permalink:: http://dlmf.nist.gov/36.10.E10
Encodings:: TeX, pMML, png

§36.10(iii) Operator Equations

Notes:: Use repeated differentiations with respect to or , in combinations that generate exact derivatives of the exponents in (36.2.5).
Keywords:: differential equations, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.10.iii

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

36.10.11

${\mathop{\Phi_{{s}}\/}\nolimits^{{(\mathrm{U})}}}\left(-i\frac{\partial}{% \partial x},-i\frac{\partial}{\partial y};\mathbf{x}\right)\mathop{\Psi^{{(% \mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)=0,$

${\mathop{\Phi_{{t}}\/}\nolimits^{{(\mathrm{U})}}}\left(-i\frac{\partial}{% \partial x},-i\frac{\partial}{\partial y};\mathbf{x}\right)\mathop{\Psi^{{(% \mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)=0,$

Symbols:: $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical umbilic integral, $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : variable, : variable and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

where

36.10.12

${\mathop{\Phi_{{s}}\/}\nolimits^{{(\mathrm{U})}}}\!\left(s,t;\mathbf{x}\right)% =\frac{\partial}{\partial s}\mathop{\Phi^{{(\mathrm{U})}}\/}\nolimits\!\left(s% ,t;\mathbf{x}\right),$

${\mathop{\Phi_{{t}}\/}\nolimits^{{(\mathrm{U})}}}\!\left(s,t;\mathbf{x}\right)% =\frac{\partial}{\partial t}\mathop{\Phi^{{(\mathrm{U})}}\/}\nolimits\!\left(s% ,t;\mathbf{x}\right).$

Symbols:: $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : variable and : variable
Permalink:: http://dlmf.nist.gov/36.10.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

Explicitly,

36.10.13 $6\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{\partial x% \partial y}-2iz\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{% \partial y}+y\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits=0,$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\partial x$ : partial differential of , : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E13
Encodings:: TeX, pMML, png

36.10.14 $3\left(\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{% \partial x}^{2}}-\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits% }{{\partial y}^{2}}\right)+2iz\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}% \nolimits}{\partial x}-x\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits=0.$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : real parameter
Referenced by:: 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 $\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial x}$ , rather than $\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{\partial x}$ :
$3\left(\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{% \partial x}^{2}}-\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits% }{{\partial y}^{2}}\right)+2iz\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}% \nolimits}{\partial x}-x\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits=0.$

Reported 2010-04-02

36.10.15 $3\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{{\partial x}^{% 2}}+iz\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial y}-x% \mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits=0,$

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E15
Encodings:: TeX, pMML, png

36.10.16 $3\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{{\partial y}^{% 2}}+iz\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial x}-y% \mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits=0.$

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E16
Encodings:: TeX, pMML, png

§36.10(iv) Partial -Derivatives

Notes:: Use repeated differentiations with respect to , or , in combinations that generate exact derivatives of the exponents in (36.2.5).
Keywords:: paraxial wave equation
Permalink:: http://dlmf.nist.gov/36.10.iv

36.10.17 $i\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{\partial z}=\frac{{% \partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{\partial x}^{2}}+% \frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{\partial y}^{2% }},$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, : real parameter and : real parameter
Referenced by:: §36.10(iv)
Permalink:: http://dlmf.nist.gov/36.10.E17
Encodings:: TeX, pMML, png

36.10.18 $i\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial z}=\frac{{% \partial}^{2}\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial x\partial y}.$

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\partial x$ : partial differential of , : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.10.E18
Encodings:: TeX, pMML, png

Equation (36.10.17) is the paraxial wave equation.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 36.9 Integral Identities 36.11 Leading-Order Asymptotics

§36.10 Differential Equations

Contents

§36.10(i) Equations for

¶ Special Cases

§36.10(ii) Partial Derivatives with Respect to the

¶ Special Cases

§36.10(iii) Operator Equations

§36.10(iv) Partial -Derivatives

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

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