# On the Approximate Evaluation of Some Oscillatory Integrals

^{*}

## Abstract

**:**

## 1. Introduction

**a**it is possible that some higher derivatives are equal to 0 at the saddle points, which causes coalescence of these points, increases the value of the integral that manifests itself as “rainbow and glory” [1]. To analyze the rainbow phenomenon, it is important to know the caustic surface $C({\mathbf{a}}_{\mathbf{c}})$, defined by the relation ${\scriptscriptstyle \frac{\partial f({\mathbf{a}}_{c};{u}_{\mathbf{s}})}{\partial {u}_{1s}}}={\scriptscriptstyle \frac{{\partial}^{2}\partial f({\mathbf{a}}_{c};{u}_{\mathbf{s}})}{\partial {u}_{1s}{}^{2}}}=\dots ={\scriptscriptstyle \frac{\partial \partial f({\mathbf{a}}_{c};{u}_{\mathbf{s}})}{\partial {u}_{ks}}}={\scriptscriptstyle \frac{{\partial}^{2}\partial f({\mathbf{a}}_{c};{u}_{\mathbf{s}})}{\partial {u}_{ks}{}^{2}}}=0$.

**a**= {a

_{1}, a

_{2}, …} is a set of parameters. As

**a**varies, as many as K + 1 (real or complex) critical points of the smooth, real-valued phase function f can coalesce in clusters of two or more. The function g has a smooth amplitude. In what follows we denote ${\scriptscriptstyle \frac{{\partial}^{n}}{\partial {u}^{n}}}f(\mathbf{a};u)={f}^{(n)}(\mathbf{a};u)$. The critical (stationary) points u

_{j}(

**a**), 1 ≤ j ≤ K + 1, are defined by ${f}^{(1)}(\mathbf{a};{u}_{j})=0$ [5].

_{max}($1\le {j}_{\mathrm{max}}\le K$) isolated real critical points [7]. The main contribution to the integral comes from the regions around the stationary points ${u}_{j}$ where the phase function $f(\mathbf{a};u)$ is slowly varying.

**a**, they can move close together and coalesce as

**a**varies. In the uniform asymptotic evaluation of oscillatory integrals the result is expressed in terms of certain canonical integrals [5,7] and their derivatives. Each canonical integral is characterized by a given number of coalescing critical points. One defines a mapping u(

**a**;t) by relating f(

**a**;u) to the normal form of cuspoid catastrophes ${\Phi}_{K}(\mathbf{b};t)$ in the following way:

**a**) and

**b**(

**a**) determined by the correspondence of K + 1 critical points of f and Φ

_{K}.

**a**the stationary points are real and ${u}_{1}\le {u}_{e}\le {u}_{2}$. For $\mathbf{a}={\mathbf{a}}_{e}$ the two points coalesce and ${u}_{1}={u}_{e}={u}_{2}$. For other values of

**a**the stationary points are complex conjugate solutions of Equation (2), i.e., ${u}_{1}={u}_{2}^{*}$.

**a**) and

**b**(

**a**), a set of nonlinear equations has to be solved. These can be solved in principle, but there are practical difficulties in attempting a solution [10]. On the other hand, away from

**b**= 0 the canonical integrals can be approximated in terms of canonical integrals ${\Phi}_{J}$ corresponding to lower-order catastrophes (i.e., J < K) [7,10,11,12,13,14,15,16].

## 2. A New Procedure for Approximate Evaluation of Oscillatory Integrals

## 3. Results

#### 3.1. Cusp Catastrophe (K = 2)

#### 3.1.1. Case $x>0$

#### 3.1.2. Case $x<0$

#### 3.2. Swallow-Tail Catastrophe (K = 3)

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Adam, J.A. The mathematical physics of rainbows and glories. Phys. Rep.
**2002**, 356, 229–365. [Google Scholar] [CrossRef] - Thom, R. Topological models in biology. Topology
**1969**, 8, 313–335. [Google Scholar] [CrossRef] - Thom, R. Stabilité Structurelle et Morphogénèse. Essai d’une Théorie Générale des Modèles; Benjamin: New York, NY, USA, 1971; p. 384. [Google Scholar]
- Connor, J.N.L. Catastrophes and molecular collisions. Mol. Phys.
**1976**, 31, 33–55. [Google Scholar] [CrossRef] - Berry, M.V.; Howls, C.J. Integrals with coalescing saddles. In NIST Handbook of Mathematical Functions; Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W., Eds.; Cambridge University Press: Cambridge, UK, 2010; Chapter 36; pp. 775–793. [Google Scholar]
- Bleistein, N.; Handelsman, R.A. Asymptotic Expansions of Integrals; Dover Publications Inc.: New York, NY, USA, 1986. [Google Scholar]
- Connor, J.N.L. Semiclassical theory of molecular collisions: Many nearly coincident classical trajectories. Mol. Phys.
**1974**, 27, 853–866. [Google Scholar] [CrossRef] - Connor, J.N.L.; Marcus, R.A. Theory of Semiclassical Transition Probabilities for Inelastic and Reactive Collisions. II Asymptotic Evaluation of the S Matrix. J. Chem. Phys.
**1971**, 55, 5636–5643. [Google Scholar] [CrossRef] - Beuc, R.; Horvatić, B.; Movre, M. Semiclassical description of collisionaly induced rainbow satellites: A model study. J. Phys. B At. Mol. Opt. Phys.
**2010**, 43, 215210. [Google Scholar] [CrossRef] - Connor, J.N.L.; Curtis, P.R.; Farrelly, D. The uniform asymptotic swallowtail approximation: Practical methods for oscillating integrals with four coalescing saddle points. J. Phys. A Math. Gen.
**1984**, 17, 283–310. [Google Scholar] [CrossRef] - Beuc, R.; Movre, M.; Pichler, G. High Temperature Optical Spectra of Diatomic Molecules at Local Thermodynamic Equilibrium. Atoms
**2018**, 6, 67. [Google Scholar] [CrossRef] - Connor, J.N.L. Semiclassical theory of molecular collisions: Three nearly coincident classical trajectories. Mol. Phys.
**1973**, 26, 1217–1231. [Google Scholar] [CrossRef] - Connor, J.N.L.; Farrelly, D. Theory of cusped rainbows in elastic scattering: Uniform semiclassical calculations using Pearcey’s integral. J. Chem. Phys.
**1981**, 75, 2831–2846. [Google Scholar] [CrossRef] - Stamnes, J.J.; Spjelkavik, B. Evaluation of the field near a cusp of a caustic. Opt. Acta
**1983**, 30, 1331–1358. [Google Scholar] [CrossRef] - Paris, R.B. The asymptotic behaviour of Pearcey’s integral for complex variables. Proc. R. Soc. Lond. A
**1991**, 432, 391–426. [Google Scholar] [CrossRef] - Kaminski, D. Asymptotic expansion of the Pearcey integral near the caustic. SIAM J. Math. Anal.
**1989**, 20, 987–1005. [Google Scholar] [CrossRef] - Chester, C.; Friedman, B.; Ursell, F. An extension of the method of steepest descents. Proc. Camb. Philos. Soc.
**1957**, 53, 599–611. [Google Scholar] [CrossRef] - Hobbs, C.A.; Connor, J.N.L.; Kirk, N.P. Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions. J. Comput. Appl. Math.
**2007**, 207, 192–213. [Google Scholar] [CrossRef] - Krüger, H. Semiclassical bound-continuum Franck-Condon factors uniformly valid at 4 coinciding critical points: 2 Crossings and 2 turning points. Theor. Chim. Acta
**1981**, 59, 97–116. [Google Scholar] [CrossRef] - Rojas, D.H.; Krüger, H. Uniform asymptotic approximations to the Franck-Condon factors. J. Phys. B At. Mol. Phys.
**1986**, 19, 1553–1575. [Google Scholar] [CrossRef]

**Figure 1.**Saddle points (${{u}_{1}}$, ${{u}_{2}}$, and ${{u}_{3}}$) of the function ${f}^{(1)}(1,y;u)$.

**Figure 2.**Function $\left|P(x,y)\right|$ (

**a**), function $\left|{P}_{q}(x,y)\right|$ (

**b**) and the relative difference $\left|\frac{{P}_{q}(x,y)-P(x,y)}{P(x,y)}\right|$ (

**c**) for $x\in \left[0,10\right]$ and $y\in \left[0,10\right]$.

**Figure 3.**Saddle points (${{u}_{1}}$, ${{u}_{2}}$, and ${{u}_{3}}$) of the functions ${f}^{(1)}(-1,y;u)$ and ${f}_{1}^{(1)}(-1,y;u)$ are shown in (

**a**,

**b**), respectively.

**Figure 4.**Function $\left|P(x,y)\right|$ (

**a**), function $\left|{P}_{A}(x,y)\right|$ (

**b**) and the absolute value of the functions’ difference $\Delta P=\left|{P}_{A}(x,y)-P(x,y)\right|$ (

**c**) for $x\in \left[-10,0\right]$ and $y\in \left[0,10\right]$.

**Figure 5.**Saddle points (${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, and ${{u}_{4}}$) of the functions ${f}^{(1)}(-1,0,z;u)$ (

**a**), ${f}_{1}^{(1)}(-1,0,z;u)$ (

**b**), and ${f}_{2}^{(1)}(-1,0,z;u)$ (

**c**).

**Figure 6.**Function $S(x,0,z)$ (

**a**), function ${S}_{A}(x,0,z)$ (

**b**) and the difference of the functions $\left|{S}_{A}(x,0,z)-S(x,0,z)\right|$ (

**c**) for $x\in \left[10,0\right]$ and $z\in \left[-10,10\right]$.

$\begin{array}{ccc}i=1& -\infty \le u<{u}_{p,1}& u\ge {u}_{p,1}\\ {f}_{1}(\mathbf{a};u)=& f(\mathbf{a};u)& {f}_{p,1}(\mathbf{a};u)\\ {g}_{1}(u)=& g(u)& g({u}_{p,1})\end{array}$ | $\begin{array}{ccc}i=m+1& u<{u}_{p,m}& {u}_{p,m}\le u\le \infty \\ {f}_{m+1}(\mathbf{a};u)=& {f}_{p,m}(\mathbf{a};u)& f(\mathbf{a};u)\\ {g}_{m+1}(u)=& g({u}_{p,m})& g(u)\end{array}$ |

$\begin{array}{cccc}1<i<m+1& u<{u}_{p,i-1}& {u}_{p,i-1}\le u<{u}_{p,i}& u\ge {u}_{p,i}\\ {f}_{i}(\mathbf{a};u)=& {f}_{p,i-1}(\mathbf{a};u)& f(\mathbf{a};u)& {f}_{p,i}(\mathbf{a};u)\\ {g}_{i}(u)=& g({u}_{p,i-1})& g(u)& g({u}_{p,i})\end{array}$ |

x | P(x,0) | P_{q}(x,0) | y | P(0,y) | P_{q}(0,y) |
---|---|---|---|---|---|

1 | 1.20838 + 0.779288i | 1.25331 + 1.25331i | 1 | 1.55093 + 0.427893i | 1.09286 + 0.353605i |

2 | 0.924029 + 0.729007i | 0.886227 + 0.886227i | 2 | 1.12475 − 0.17608i | 0.837873 − 0.359347i |

3 | 0.754294 + 0.657362i | 0.723601 + 0.723601i | 3 | 0.384485 − 0.642953i | 0.244423 − 0.757992i |

4 | 0.646979 + 0.593695i | 0.626657 + 0.626657i | 4 | −0.385925− 0.545144i | −0.434337 − 0.578749i |

5 | 0.573931 + 0.541858i | 0.560499 + 0.560499i | 5 | −0.670195 + 0.0710806i | −0.66748 + 0.0754601i |

6 | 0.520848 + 0.500054i | 0.511663 + 0.511663i | 6 | −0.235367 + 0.592027i | −0.214718 + 0.594539i |

7 | 0.480234 + 0.465936i | 0.473708 + 0.473708i | 7 | 0.430078 + 0.415615i | 0.442629 + 0.405754i |

8 | 0.447916 + 0.437618i | 0.443113 + 0.443113i | 8 | 0.510179 − 0.260969i | 0.506489 − 0.270769i |

9 | 0.421413 + 0.413718i | 0.417771 + 0.417771i | 9 | −0.1039 − 0.542069i | −0.112749 − 0.540578i |

10 | 0.399166 + 0.393242i | 0.396333 + 0.396333i | 10 | −0.532222 − 0.0242518i | −0.532896 − 0.0165838i |

**Table 3.**Values of P(x,y) obtained for large negative values of x when y = 2 and 4 compared to the asymptotic values [15] and the present work.

x | y | P(x,y) | Asymptotic [11] | P_{A}(x,y) |
---|---|---|---|---|

−4 | 2 | 1.96341 − 0.73419i | 1.97363 − 0.72605i | 1.96482 − 0.72731i |

−6 | 2 | 0.96527 + 0.46413i | 0.96537 + 0.46415i | 0.96366 + 0.46538i |

−8 | 2 | 1.00422 − 0.11480i | 1.00422 − 0.11480i | 1.004077 −0.11392i |

−4 | 4 | 0.14360 + 0.90244i | − | 0.14063 + 0.90013i |

−6 | 4 | 0.29478 − 0.84373i | 0.29399 − 0.84356i | 0.29629 − 0.84406i |

−8 | 4 | 0.75372 − 0.23933i | 0.75371 − 0.23933i | 0.75379 − 0.23889i |

x | y | P(x,y) | Kaminski [10] | P_{A}(x,y) |
---|---|---|---|---|

−1.0 | 0.544331 | 2.14158 + 0.0990191i | 2.34415 + 0.00118008i | 2.1003 + 0.156994i |

−2.0 | 1.5396 | 0.962205 − 0.450083i | 0.926925 − 0.428207i | 0.965935 − 0.448303i |

−3.0 | 2.82843 | 1.13215 + 1.19182i | 1.14743 + 1.19594i | 1.12358 + 1.19408i |

−4.0 | 4.35465 | −0.142478 + 1.20972i | −0.143582 + 1.2217i | −0.146125 + 1.20649i |

−5.0 | 6.08581 | −0.888104 + 0.979074i | −0.890885 + 0.983784i | −0.889185 + 0.975844i |

−6.0 | 8. | −1.10157 + 0.582286i | −1.09951 + 0.581515i | −1.1015 + 0.58047i |

−7.0 | 10.0812 | −0.249906 − 0.91133i | −0.249866 − 0.914663i | −0.248282 − 0.910954i |

−8.0 | 12.3168 | 0.321769 − 0.468203i | 0.324275 − 0.466919i | 0.321939 − 0.467325i |

−9.0 | 14.6969 | 0.495502 + 0.309572i | 0.495034 + 0.311661i | 0.494746 + 0.309898i |

−10.0 | 17.2133 | −0.704129 + 0.779039i | −0.704954 + 0.779772i | −0.70467 + 0.778148i |

**Table 5.**Comparison of the values of $S(x,0,z)$ and ${S}_{A}(x,0,z)$ on the caustics $z=0$ and $z={\scriptscriptstyle \frac{9}{20}}{x}^{2}$.

x | z | S(x,0,0) | S_{A}(x,0,0) | z = (9x^{2})/20 | S(x,0,z) | S_{A}(x,0,z) |
---|---|---|---|---|---|---|

−1 | 0. | 2.269084 | 1.913266 | 0.45 | 2.043860 | 1.661334 |

−2 | 0. | 2.409949 | 2.394455 | 1.8 | 1.663543 | 1.612423 |

−3 | 0. | 0.596406 | 0.604199 | 4.05 | −0.519597 | −0.508896 |

−4 | 0. | 1.292366 | 1.281152 | 7.2 | −0.915491 | −0.908131 |

−5 | 0. | 0.215598 | 0.212031 | 11.25 | 0.821846 | 0.816979 |

−6 | 0. | 0.247304 | 0.245451 | 16.2 | 0.660728 | 0.660188 |

−7 | 0. | 0.670358 | 0.667798 | 22.05 | −0.394882 | −0.392698 |

−8 | 0. | 0.834808 | 0.83448 | 28.8 | 0.204247 | 0.205210 |

−9 | 0. | 1.155092 | 1.154554 | 36.45 | 1.055443 | 1.053662 |

−10 | 0. | 0.767679 | 0.767563 | 45. | −0.891812 | −0.890310 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Beuc, R.; Movre, M.; Horvatić, B.
On the Approximate Evaluation of Some Oscillatory Integrals. *Atoms* **2019**, *7*, 47.
https://doi.org/10.3390/atoms7020047

**AMA Style**

Beuc R, Movre M, Horvatić B.
On the Approximate Evaluation of Some Oscillatory Integrals. *Atoms*. 2019; 7(2):47.
https://doi.org/10.3390/atoms7020047

**Chicago/Turabian Style**

Beuc, Robert, Mladen Movre, and Berislav Horvatić.
2019. "On the Approximate Evaluation of Some Oscillatory Integrals" *Atoms* 7, no. 2: 47.
https://doi.org/10.3390/atoms7020047