Geodesic
Approximate formulas for moderately small eikonal amplitudes
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 2, pp. 273-287 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions $J_0(z)$. We generalize the expression with four functions $J_0(z)$ and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions $J_\nu(az)$ to the case with noninteger $\nu$ and complex $a$.
Keywords: scattering amplitude, eikonal approximation, improper integral, Bessel function. 
@article{TMF_2016_188_2_a4,
     author = {A. V. Kisselev},
     title = {Approximate formulas for moderately small eikonal amplitudes},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {273--287},
     year = {2016},
     volume = {188},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a4/}
}
TY  - JOUR
AU  - A. V. Kisselev
TI  - Approximate formulas for moderately small eikonal amplitudes
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 273
EP  - 287
VL  - 188
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a4/
LA  - ru
ID  - TMF_2016_188_2_a4
ER  - 
%0 Journal Article
%A A. V. Kisselev
%T Approximate formulas for moderately small eikonal amplitudes
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 273-287
%V 188
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a4/
%G ru
%F TMF_2016_188_2_a4
A. V. Kisselev. Approximate formulas for moderately small eikonal amplitudes. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 2, pp. 273-287. http://geodesic.mathdoc.fr/item/TMF_2016_188_2_a4/

[1] G. Molière, Z. Naturforsch. A, 2:3 (1947), 133–145 ; Z. Naturforsch. A, 3:2 (1948), 78–97 | DOI | Zbl | DOI

[2] R. J. Glauber, “High-energy collision theory”, Lectures in Theoretical Physics (University of Colorado, Boulder, 1958), v. 1, eds. W. E. Brittin, L. G. Dunham, Interscience, New York, 1959, 315–414 | MR

[3] R. G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill, New York, 1966 | DOI | MR

[4] H. Cheng, T. T. Wu, Phys. Rev. Lett., 22:13 (1969), 666–669 | DOI

[5] A. A. Logunov, A. N. Tavkhelidze, Nuovo Cimento, 29:2 (1963), 380–389 | DOI | MR

[6] V. R. Garsevanishvili, V. A. Matveev, L. A. Slepchenko, A. N. Tavkhelidze, Phys. Lett. B, 29:3 (1969), 191–192 | DOI

[7] V. A. Petrov, “High-energy implications of extended unitarity”, Proceedings of 7th Conference (Blois Workshop) on Elastic and Diffractive Scattering (Blois, France, June 20–24, 1995), eds. P. Chiappetta, M. Haguenauer, J. Trân Thanh Vân, Ed. Frontières, Gif-sur-Yvette, 1996, 139–143

[8] A. V. Kisselev, V. A. Petrov, Eur. Phys. J. C, 36:1 (2004), 103-111, arXiv: hep-ph/0311356 | DOI

[9] S. K. Lucas, J. Comput. Appl. Math., 64:3 (1995), 269–282 | DOI | MR | Zbl

[10] J. Van Deun, R. Cools, Comp. Phys. Commun., 178:8 (2008), 578–590 | DOI | MR | Zbl

[11] J. T. Conway, Mon. Not. R. Astron. Soc., 316:3 (2000), 540–554 | DOI

[12] R. Mehrem, J. T. Londergan, M. H. Macfarlane, J. Phys. A: Math. Gen., 24:7 (1991), 1435–1453 | DOI | MR | Zbl

[13] A. V. Kisselev, The Open Astronomy J., 2:1 (2009), 12–15 | DOI

[14] J. D. Jackson, Classical Electrodynamics, 3rd ed., John Wiley and Sons, New York, 1998 | MR

[15] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963 | MR | Zbl

[16] G. N. Vatson, Teoriya besselevykh funktsii, IL, M., 1949 | MR | Zbl

[17] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii, v. 1, Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1959 | MR

[18] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady, v. 2, Spetsialnye funktsii, Nauka, M., 1983 | MR | MR | Zbl

[19] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 3, Ellipticheskie i avtomorfnye funktsii, funktsii Lame i Mate, Nauka, M., 1967 | MR | Zbl

[20] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974 | MR | MR | Zbl