# §36.12 Uniform Approximation of Integrals

## §36.12(i) General Theory for Cuspoids

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 $I(\mathbf{y},k)=\int_{-\infty}^{\infty}\exp\left(ikf(u;\mathbf{y})\right)g(u,% \mathbf{y})\mathrm{d}u,$ ⓘ Defines: $I(\mathbf{y},k)$: oscillatory integral (locally) Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\int$: integral, $k$: variable, $g(u,\mathbf{y})$: 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 See also: Annotations for §36.12(i), §36.12 and Ch.36

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

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

The leading-order uniform asymptotic approximation is given by

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

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 $\Phi_{K}\left(t;\mathbf{x}\right)$ in the following way:

 36.12.4 $f(u(t,\mathbf{y});\mathbf{y})=A(\mathbf{y})+\Phi_{K}\left(t;\mathbf{x}(\mathbf% {y})\right),$ ⓘ Defines: $u(t;\mathbf{y})$: mapping (locally) Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$: cuspoid catastrophe of codimension $K$, $t$: variable, $K$: codimension, $f(u,\mathbf{y})$: function and $A(\mathbf{y})$: function Permalink: http://dlmf.nist.gov/36.12.E4 Encodings: TeX, pMML, png See also: Annotations for §36.12(i), §36.12 and Ch.36

with the $K+1$ functions $A(\mathbf{y})$ and $\mathbf{x}(\mathbf{y})$ determined by correspondence of the $K+1$ critical points of $f$ and $\Phi_{K}$. Then

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

where $t_{j}(\mathbf{x})$, $1\leq j\leq K+1$, are the critical points of $\Phi_{K}$, 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 $A(\mathbf{y})=f(u(0,\mathbf{y});\mathbf{y}),$ ⓘ Defines: $A(\mathbf{y})$: function (locally) Symbols: $f(u,\mathbf{y})$: function and $u(t;\mathbf{y})$: mapping Permalink: http://dlmf.nist.gov/36.12.E6 Encodings: TeX, pMML, png See also: Annotations for §36.12(i), §36.12 and Ch.36
 36.12.7 $\displaystyle\mathbf{z}(\mathbf{y};k)$ $\displaystyle=\{z_{1}(\mathbf{y};k),z_{2}(\mathbf{y};k),\dots,z_{K}(\mathbf{y}% ;k)\},$ $\displaystyle z_{m}(\mathbf{y};k)$ $\displaystyle=x_{m}(\mathbf{y})k^{1-(m/(K+2))},$ ⓘ Defines: $\mathbf{z}(\mathbf{y},k)$: function (locally) and $z_{m}(\mathbf{y},k)$: function (locally) Symbols: $n$: integer, $k$: variable, $K$: codimension and $\mathbf{x}(\mathbf{y})$: function Permalink: http://dlmf.nist.gov/36.12.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.12(i), §36.12 and Ch.36
 36.12.8 $a_{m}(\mathbf{y})=\sum_{n=1}^{K+1}\frac{P_{mn}(\mathbf{y})G_{n}(\mathbf{y})}{(% t_{n}(\mathbf{x}(\mathbf{y})))^{m+1}\prod\limits_{\begin{subarray}{c}l=1\\ l\neq n\end{subarray}}^{K+1}(t_{n}(\mathbf{x}(\mathbf{y}))-t_{l}(\mathbf{x}(% \mathbf{y})))},$ ⓘ Defines: $a_{m}(\mathbf{y})$: function (locally) Symbols: $l$: integer, $n$: integer, $m$: integer, $K$: codimension and $t_{j}(\mathbf{x})$: solutions Permalink: http://dlmf.nist.gov/36.12.E8 Encodings: TeX, pMML, png See also: Annotations for §36.12(i), §36.12 and Ch.36

where

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

and

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

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 $g$ is real analytic.

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

For further information see Berry and Howls (1993).

## §36.12(ii) Special Case

For $K=1$, with a single parameter $y$, let the two critical points of $f(u;y)$ be denoted by $u_{\pm}(y)$, with $u_{+}>u_{-}$ for those values of $y$ for which these critical points are real. Then

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

where

 36.12.12 $\displaystyle\widetilde{f}$ $\displaystyle=\tfrac{1}{2}(f(u_{+}(y),y)+f(u_{-}(y),y)),$ $\displaystyle g_{\pm}$ $\displaystyle=g(u_{\pm}(y),y),$ $\displaystyle f_{\pm}^{\prime\prime}$ $\displaystyle=\frac{{\partial}^{2}}{{\partial u}^{2}}f(u_{\pm}(y),y),$ $\displaystyle\Delta$ $\displaystyle=\left(\tfrac{3}{4}(f(u_{-}(y),y)-f(u_{+}(y),y))\right)^{2/3}.$

For $\mathrm{Ai}$ and $\mathrm{Ai}'$ 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 $\mathrm{Ai}$ and $\mathrm{Ai}'$ are real if $y$ is real and $g$ 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 $y$ is such that both critical points are real, and by analytic continuation otherwise.