Abstract
We have applied the method of stationary phase to evaluate double and multiple integrals of the type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) for large values of the parameter k. In the first part we have established in a rigorous manner the stationary phase method to double and multiple integrals of type (A). Furthermore we have obtained an asymptotic expansion of (A), if the amplitude and phase functions can be developed in a canonical form near the vicinity of critical or stationary points of the integral. This development contains as particular cases all those which are important in physical applications, especially, to diffraction and scattering of electromagnetic and corpuscular waves by optical systems, diffracting bodies and potential scatterers. In the second part we have considered the problem of convergence of the expansion of the principal contribution to the integral in the asymptotic sense of Poincare. The proof is based on the increasing method used in mathematical analysis. The third part is devoted to the derivation of various asymptotic series due to different types of critical or stationary points associated with the amplitude and phase functions. In the fourth part we have generalized the method to
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Chako, N
[1]
- Commissariat a l'Energie Atomique, Saclay (France). Centre d'Etudes Nucleaires
Citation Formats
Chako, N.
Study of the asymptotic expansion of multiple integrals in mathematical physics; Etudes sur les developpements asymptotiques des integrales multiples de la physique mathematique.
France: N. p.,
1968.
Web.
Chako, N.
Study of the asymptotic expansion of multiple integrals in mathematical physics; Etudes sur les developpements asymptotiques des integrales multiples de la physique mathematique.
France.
Chako, N.
1968.
"Study of the asymptotic expansion of multiple integrals in mathematical physics; Etudes sur les developpements asymptotiques des integrales multiples de la physique mathematique."
France.
@misc{etde_20723426,
title = {Study of the asymptotic expansion of multiple integrals in mathematical physics; Etudes sur les developpements asymptotiques des integrales multiples de la physique mathematique}
author = {Chako, N}
abstractNote = {We have applied the method of stationary phase to evaluate double and multiple integrals of the type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) for large values of the parameter k. In the first part we have established in a rigorous manner the stationary phase method to double and multiple integrals of type (A). Furthermore we have obtained an asymptotic expansion of (A), if the amplitude and phase functions can be developed in a canonical form near the vicinity of critical or stationary points of the integral. This development contains as particular cases all those which are important in physical applications, especially, to diffraction and scattering of electromagnetic and corpuscular waves by optical systems, diffracting bodies and potential scatterers. In the second part we have considered the problem of convergence of the expansion of the principal contribution to the integral in the asymptotic sense of Poincare. The proof is based on the increasing method used in mathematical analysis. The third part is devoted to the derivation of various asymptotic series due to different types of critical or stationary points associated with the amplitude and phase functions. In the fourth part we have generalized the method to multiple integrals and to the case where the parameter k enter implicitly in the phase function The latter type of integrals extend the scope of the former type to a number of important physical problems; for instance, to the propagation of waves in dispersive and absorbing media. In the last chapter we have made a study and compared the results obtained by the application of the stationary phase method to the integrals (double) of diffraction and the results derived by using the Young-Rubinowicz method. Result of our analysis shows the equivalence of the two methods of approach to the problems of diffraction based, on one hand, on the Fresnel-Kirchhoff theory and, on the other hand, the Young-Rubinowicz theory, provided one interprets in a proper manner the results derived from the two methods, especially the expression of the geometrical wave. (author) [French] Nous avons applique la methode de la phase stationnaire pour evaluer les integrales doubles et multiples du type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) pour les grandes valeurs du parametre k. Dans la premiere partie nous avons etendu d'une maniere rigoureuse la methode de la phase stationnaire aux integrales doubles et multiples de type (A). De plus, nous avons obtenu un developpement asymptotique de (A), lorsque l'amplitude et la phase peuvent se developper sous forme canonique au voisinage de points critiques ou stationnaires de l'integrale (A). Ce developpement contient comme cas particuliers tous les cas importants dans les applications physiques et particulierement en diffraction et diffusion d'ondes electromagnetiques et corpusculaires par des systemes optiques, corps diffractants et potentiels de diffusions. Dans la seconde partie nous avons considere le probleme de la convergence du developpement de la contribution principale a l'integrale, au sens asymptotique de Poincare. La preuve est basee sur la methode des majorantes, utilisee en analyse mathematique. La troisieme partie contient la derivation des series asymptotiques diverses, due aux types varies de points critiques ou stationnaires lies aux fonctions d'amplitude et de phase. Dans la quatrieme partie nous avons generalise la methode aux integrales multiples et au cas ou le parametre k entre implicitement dans la fonction de phase. Ce dernier type d'integrales permet l'extension du premier type a de nombreux problemes physiques, par exemple a la propagation d'ondes en milieux dispersifs et absorbants. Au dernier chapitre, nous faisons l'etude des integrales doubles de diffractions (theorie de Kirchhoff) et nous comparons les resultats par l'application de la methode de la phase stationnaire et de la methode Young-Rubinowicz. Le resultat de l'analyse montre l'equivalence des deux methodes d'approche aux problemes de diffraction basees, l'une sur la theorie Fresnel-Kirchhoff, l'autre sur celle de Young-Rubinowicz, a condition d'etre prudent dans l'interpretation des resultats obtenus par les deux methodes. (auteur)}
place = {France}
year = {1968}
month = {Jul}
}
title = {Study of the asymptotic expansion of multiple integrals in mathematical physics; Etudes sur les developpements asymptotiques des integrales multiples de la physique mathematique}
author = {Chako, N}
abstractNote = {We have applied the method of stationary phase to evaluate double and multiple integrals of the type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) for large values of the parameter k. In the first part we have established in a rigorous manner the stationary phase method to double and multiple integrals of type (A). Furthermore we have obtained an asymptotic expansion of (A), if the amplitude and phase functions can be developed in a canonical form near the vicinity of critical or stationary points of the integral. This development contains as particular cases all those which are important in physical applications, especially, to diffraction and scattering of electromagnetic and corpuscular waves by optical systems, diffracting bodies and potential scatterers. In the second part we have considered the problem of convergence of the expansion of the principal contribution to the integral in the asymptotic sense of Poincare. The proof is based on the increasing method used in mathematical analysis. The third part is devoted to the derivation of various asymptotic series due to different types of critical or stationary points associated with the amplitude and phase functions. In the fourth part we have generalized the method to multiple integrals and to the case where the parameter k enter implicitly in the phase function The latter type of integrals extend the scope of the former type to a number of important physical problems; for instance, to the propagation of waves in dispersive and absorbing media. In the last chapter we have made a study and compared the results obtained by the application of the stationary phase method to the integrals (double) of diffraction and the results derived by using the Young-Rubinowicz method. Result of our analysis shows the equivalence of the two methods of approach to the problems of diffraction based, on one hand, on the Fresnel-Kirchhoff theory and, on the other hand, the Young-Rubinowicz theory, provided one interprets in a proper manner the results derived from the two methods, especially the expression of the geometrical wave. (author) [French] Nous avons applique la methode de la phase stationnaire pour evaluer les integrales doubles et multiples du type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) pour les grandes valeurs du parametre k. Dans la premiere partie nous avons etendu d'une maniere rigoureuse la methode de la phase stationnaire aux integrales doubles et multiples de type (A). De plus, nous avons obtenu un developpement asymptotique de (A), lorsque l'amplitude et la phase peuvent se developper sous forme canonique au voisinage de points critiques ou stationnaires de l'integrale (A). Ce developpement contient comme cas particuliers tous les cas importants dans les applications physiques et particulierement en diffraction et diffusion d'ondes electromagnetiques et corpusculaires par des systemes optiques, corps diffractants et potentiels de diffusions. Dans la seconde partie nous avons considere le probleme de la convergence du developpement de la contribution principale a l'integrale, au sens asymptotique de Poincare. La preuve est basee sur la methode des majorantes, utilisee en analyse mathematique. La troisieme partie contient la derivation des series asymptotiques diverses, due aux types varies de points critiques ou stationnaires lies aux fonctions d'amplitude et de phase. Dans la quatrieme partie nous avons generalise la methode aux integrales multiples et au cas ou le parametre k entre implicitement dans la fonction de phase. Ce dernier type d'integrales permet l'extension du premier type a de nombreux problemes physiques, par exemple a la propagation d'ondes en milieux dispersifs et absorbants. Au dernier chapitre, nous faisons l'etude des integrales doubles de diffractions (theorie de Kirchhoff) et nous comparons les resultats par l'application de la methode de la phase stationnaire et de la methode Young-Rubinowicz. Le resultat de l'analyse montre l'equivalence des deux methodes d'approche aux problemes de diffraction basees, l'une sur la theorie Fresnel-Kirchhoff, l'autre sur celle de Young-Rubinowicz, a condition d'etre prudent dans l'interpretation des resultats obtenus par les deux methodes. (auteur)}
place = {France}
year = {1968}
month = {Jul}
}