36 Integrals with Coalescing Saddles Applications 36.11 Leading-Order Asymptotics 36.13 Kelvin’s Ship-Wave Pattern

§36.12 Uniform Approximation of Integrals

Keywords:: canonical integrals, integrals, saddle points
Referenced by:: §2.4(vi), §36.15(ii)
Permalink:: http://dlmf.nist.gov/36.12

§36.12(i) General Theory for Cuspoids
§36.12(ii) Special Case
§36.12(iii) Additional References

§36.12(i) General Theory for Cuspoids

Keywords:: coalescing saddle points, critical points, diffraction catastrophes
Permalink:: http://dlmf.nist.gov/36.12.i

The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. In the cuspoid case (one integration variable)

36.12.1 $I(\mathbf{y},k)=\int_{{-\infty}}^{\infty}\mathop{\exp\/}\nolimits\!\left(ikf(u% ;\mathbf{y})\right)g(u,\mathbf{y})du,$

Symbols:: : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : variable, $I(\mathbf{y},k)$ : oscillatory integral, $g(u,\mathbf{y})$ : function, $f(u,\mathbf{y})$ : function and $u(t;\mathbf{y})$ : mapping
Referenced by:: §36.13
Permalink:: http://dlmf.nist.gov/36.12.E1
Encodings:: TeX, pMML, png

where is a large real parameter and $\mathbf{y}=\{y_{1},y_{2},\dots\}$ is a set of additional (nonasymptotic) parameters. As $\mathbf{y}$ varies as many as (real or complex) critical points of the smooth phase function can coalesce in clusters of two or more. The function has a smooth amplitude. Also, is real analytic, and $\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}>0$ for all $\mathbf{y}$ such that all critical points coincide. If $\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}<0$ , then we may evaluate the complex conjugate of for real values of $\mathbf{y}$ and , and obtain by conjugation and analytic continuation. The critical points $u_{j}(\mathbf{y})$ , $1\leq j\leq K+1$ , are defined by

36.12.2 $\frac{\partial}{\partial u}f(u_{j}(\mathbf{y});\mathbf{y})=0.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $f(u,\mathbf{y})$ : function and $u(t;\mathbf{y})$ : mapping
Permalink:: http://dlmf.nist.gov/36.12.E2
Encodings:: TeX, pMML, png

The leading-order uniform asymptotic approximation is given by

36.12.3 $I(\mathbf{y},k)=\frac{\mathop{\exp\/}\nolimits\!\left(ikA(\mathbf{y})\right)}{% k^{{1/(K+2)}}}\sum\limits_{{m=0}}^{K}\frac{a_{m}(\mathbf{y})}{k^{{m/(K+2)}}}% \left(\delta_{{m,0}}-\left(1-\delta_{{m,0}}\right)i\frac{\partial}{\partial z_% {m}}\right)\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{z}(\mathbf{y};k)% \right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{k}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\delta_{{j,k}}$ : Kronecker delta, $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : variable, : codimension, $I(\mathbf{y},k)$ : oscillatory integral, $a_{m}(\mathbf{y})$ : function, $A(\mathbf{y})$ : function, $\mathbf{z}(\mathbf{y},k)$ : function and $z_{m}(\mathbf{y},k)$ : function
Permalink:: http://dlmf.nist.gov/36.12.E3
Encodings:: TeX, pMML, png

where $A(\mathbf{y})$ , $\mathbf{z}(\mathbf{y},k)$ , $a_{m}(\mathbf{y})$ are as follows. Define a mapping $u(t;\mathbf{y})$ by relating $f(u;\mathbf{y})$ to the normal form (36.2.1) of $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ in the following way:

36.12.4 $f(u(t,\mathbf{y});\mathbf{y})=A(\mathbf{y})+\mathop{\Phi_{{K}}\/}\nolimits\!% \left(t;\mathbf{x}(\mathbf{y})\right),$

Symbols:: $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, : variable, : codimension, $f(u,\mathbf{y})$ : function, $u(t;\mathbf{y})$ : mapping and $A(\mathbf{y})$ : function
Permalink:: http://dlmf.nist.gov/36.12.E4
Encodings:: TeX, pMML, png

with the functions $A(\mathbf{y})$ and $\mathbf{x}(\mathbf{y})$ determined by correspondence of the critical points of and $\mathop{\Phi_{{K}}\/}\nolimits$ . Then

36.12.5 $f(u_{j}(\mathbf{y});\mathbf{y})=A(\mathbf{y})+\mathop{\Phi_{{K}}\/}\nolimits\!% \left(t_{j}(\mathbf{x}(\mathbf{y}));\mathbf{x}(\mathbf{y})\right),$

Symbols:: $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, : codimension, $f(u,\mathbf{y})$ : function, $u(t;\mathbf{y})$ : mapping, $A(\mathbf{y})$ : function and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.12.E5
Encodings:: TeX, pMML, png

where $t_{j}(\mathbf{x})$ , $1\leq j\leq K+1$ , are the critical points of $\mathop{\Phi_{{K}}\/}\nolimits$ , that is, the solutions (real and complex) of (36.4.1). Correspondence between the $u_{j}(\mathbf{y})$ and the $t_{j}(\mathbf{x})$ is established by the order of critical points along the real axis when $\mathbf{y}$ and $\mathbf{x}$ are such that these critical points are all real, and by continuation when some or all of the critical points are complex. The branch for $\mathbf{x}(\mathbf{y})$ is such that $\mathbf{x}$ is real when $\mathbf{y}$ is real. In consequence,

36.12.6 $A(\mathbf{y})=f(u(0,\mathbf{y});\mathbf{y}),$

Symbols:: $f(u,\mathbf{y})$ : function, $u(t;\mathbf{y})$ : mapping and $A(\mathbf{y})$ : function
Permalink:: http://dlmf.nist.gov/36.12.E6
Encodings:: TeX, pMML, png

36.12.7

$\mathbf{z}(\mathbf{y};k)=\{z_{1}(\mathbf{y};k),z_{2}(\mathbf{y};k),\dots,z_{K}% (\mathbf{y};k)\},$

$z_{m}(\mathbf{y};k)=x_{m}(\mathbf{y})k^{{1-(m/(K+2))}},$

Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, : integer, : variable, : codimension, $\mathbf{x}(\mathbf{y})$ : function, $\mathbf{z}(\mathbf{y},k)$ : function and $z_{m}(\mathbf{y},k)$ : function
Permalink:: http://dlmf.nist.gov/36.12.E7
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36.12.8 $a_{m}(\mathbf{y})=\sum_{{n=1}}^{{K+1}}\frac{P_{{mn}}(\mathbf{y})G_{n}(\mathbf{% y})}{(t_{n}(\mathbf{x}(\mathbf{y})))^{{m+1}}\prod\limits_{{\substack{l=1\\ l\neq n}}}^{{K+1}}(t_{n}(\mathbf{x}(\mathbf{y}))-t_{l}(\mathbf{x}(\mathbf{y}))% )},$

Symbols:: : integer, : integer, : integer, : codimension, $a_{m}(\mathbf{y})$ : function and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.12.E8
Encodings:: TeX, pMML, png

where

36.12.9 $P_{{mn}}(\mathbf{y})={(t_{n}(\mathbf{x}(\mathbf{y})))^{{K+1}}}+\sum_{{l=m+2}}^% {K}\frac{l}{K+2}x_{l}(\mathbf{y}){(t_{n}(\mathbf{x}(\mathbf{y})))^{{l-1}}},$

Symbols:: : integer, : integer, : integer, : codimension, $\mathbf{x}(\mathbf{y})$ : function and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.12.E9
Encodings:: TeX, pMML, png

and

36.12.10 $G_{n}(\mathbf{y})=g(t_{n}(\mathbf{y}),\mathbf{y})\sqrt{\frac{\ifrac{{\partial}% ^{2}\mathop{\Phi_{{K}}\/}\nolimits\!\left(t_{n}(\mathbf{x}(\mathbf{y}));% \mathbf{x}(\mathbf{y})\right)}{{\partial t}^{2}}}{\ifrac{{\partial}^{2}f(u_{n}% (\mathbf{y}))}{{\partial u}^{2}}}}.$

Symbols:: $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : variable, : codimension, $g(u,\mathbf{y})$ : function, $f(u,\mathbf{y})$ : function, $u(t;\mathbf{y})$ : mapping and $t_{j}(\mathbf{x})$ : solutions
Referenced by:: §36.12(i)
Permalink:: http://dlmf.nist.gov/36.12.E10
Encodings:: TeX, pMML, png

In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. The square roots are real and positive when $\mathbf{y}$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. The quantities $a_{m}(\mathbf{y})$ are real for real $\mathbf{y}$ when is real analytic.

This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes $\mathop{\Psi_{{K}}\/}\nolimits\!(\mathbf{x};k)$ in (36.2.10) away from $\mathbf{x}=0$ , in terms of canonical integrals $\mathop{\Psi_{{J}}\/}\nolimits\!\left(\xi(\mathbf{x};k)\right)$ for . For example, the diffraction catastrophe $\mathop{\Psi_{{2}}\/}\nolimits\!(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function $\mathop{\Psi_{{1}}\/}\nolimits\!\left(\xi(x,y;k)\right)$ when is large, provided that and are not small. For details of this example, see Paris (1991).

For further information see Berry and Howls (1993).

§36.12(ii) Special Case

Notes:: See Berry and Howls (1993).
Referenced by:: §2.8(iii)
Permalink:: http://dlmf.nist.gov/36.12.ii

For , with a single parameter , let the two critical points of be denoted by $u_{{\pm}}(y)$ , with $u_{{+}}>u_{{-}}$ for those values of for which these critical points are real. Then

36.12.11 $I(y,k)=\frac{\Delta^{{1/4}}\pi\sqrt{2}}{k^{{1/3}}}\mathop{\exp\/}\nolimits\!% \left(ik\widetilde{f}\right)\left(\left(\frac{g_{{+}}}{\sqrt{f_{{+}}^{{\prime% \prime}}}}+\frac{g_{{-}}}{\sqrt{-f_{{-}}^{{\prime\prime}}}}\right)\mathop{% \mathrm{Ai}\/}\nolimits\!\left(-k^{{2/3}}\Delta\right)\left(1+\mathop{O\/}% \nolimits\!\left(\frac{1}{k}\right)\right)-i\left(\frac{g_{{+}}}{\sqrt{f_{{+}}% ^{{\prime\prime}}}}-\frac{g_{{-}}}{\sqrt{-f_{{-}}^{{\prime\prime}}}}\right)% \frac{{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(-k^{{2/3}}\Delta% \right)}{k^{{1/3}}\Delta^{{1/2}}}\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}% {k}\right)\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\exp\/}\nolimits z$ : exponential function, : real parameter, : variable, $I(\mathbf{y},k)$ : oscillatory integral and $f(u,\mathbf{y})$ : function
Referenced by:: §36.13
Permalink:: http://dlmf.nist.gov/36.12.E11
Encodings:: TeX, pMML, png

where

36.12.12

$\widetilde{f}=\tfrac{1}{2}(f(u_{{+}}(y),y)+f(u_{{-}}(y),y)),$

$g_{{\pm}}=g(u_{{\pm}}(y),y),$

$f_{{\pm}}^{{\prime\prime}}=\frac{{\partial}^{2}}{{\partial u}^{2}}f(u_{{\pm}}(% y),y),$

$\Delta=\left(\tfrac{3}{4}(f(u_{{-}}(y),y)-f(u_{{+}}(y),y))\right)^{{2/3}}.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real parameter, $g(u,\mathbf{y})$ : function, $f(u,\mathbf{y})$ : function and $u_{j}(\mathbf{y})$ : critical points
Referenced by:: §36.13
Permalink:: http://dlmf.nist.gov/36.12.E12
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

For $\mathop{\mathrm{Ai}\/}\nolimits$ and ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}$ see §9.2. Branches are chosen so that $\Delta$ is real and positive if the critical points are real, or real and negative if they are complex. The coefficients of $\mathop{\mathrm{Ai}\/}\nolimits$ and ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}$ are real if is real and is real analytic. Also, $\Delta^{{1/4}}/\sqrt{f_{{+}}^{{\prime\prime}}}$ and $\Delta^{{1/4}}/\sqrt{-f_{{-}}^{{\prime\prime}}}$ are chosen to be positive real when is such that both critical points are real, and by analytic continuation otherwise.

§36.12(iii) Additional References

Keywords:: Pearcey integral, canonical integrals, coalescing saddle points, critical points, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, integrals, saddle points, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.12.iii

For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

© 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. 36.11 Leading-Order Asymptotics 36.13 Kelvin’s Ship-Wave Pattern

§36.12 Uniform Approximation of Integrals

Contents

§36.12(i) General Theory for Cuspoids

§36.12(ii) Special Case

§36.12(iii) Additional References