The Use of Some Important Integrals in Physics

Abstract

The evaluation process of the two integrals ??r^m cos??kr exp(-?r)dr? and ??r^m sin??kr exp(-?r)dr? Is recapitulated , using elementary knowledge from complex variables subject. This is done in a general manner, thence some special cases are worked out in full. Once ,this is done we turn to the application of these important integrals in some physical problems. In this concern, positive energy bound states is a vital example.

Introduction

Conclusion

As we have shown in this article ;evaluating the two integrals ???r^m cos??kr exp?(-?r)dr? ? and ???r^m sin?kr exp?(-?r)dr? , was a simple task. The use of simple complex variables has made such a job simple and easy; and with no need to know the sophisticated method of the calculus of residues .Moreover, the method is elaborate, quick and instructive[1]. It should also be noted that such integrals are useful in some physical problems as we have seen in the problem of searching for a PEBS. For instance and as we have seen in section 4,if we wish to compute the scattering phase shift for a particular partial wave of order l and if the form factor is g_l=r^(l+2) exp?(-?r) ,which is the modified Yukawa potential, then we run into the problem of evaluating improper integrals such as the ones encountered in case 5. The integrals discussed and computed are just sample members of a large number of useful special integrals. To mention a few of them, we present integrals of the form I_m (x_0,a_0 )=?_0^??(x^m dx)/(1+exp?( (x-x_0)/a_0 )) (41) Such an integral is involved in computing the two- and three-parameters Fermi or Woods Saxon function, which is attained through a ceryain binomial expansion and the computation is possible because of bounded convergence and integrability [7].Moreover ,this kind of integrals are faced with in solving heavy-ion optical model potential at intermediate energies by inversion [8] The importance of some other special integrals of mathematical physics, such as probability integrals and Fresnel integrals ,and how to compute them in a fast manner is also tackled in an efficient way [9].

References

[1] Awin A. M. (1991) Some important integrals in physics ,International Journal of Mathematical Education in Science and Technology ,Vol. 22, No.1, 129-132. [2] Husain D., Awin A.M. (1984) Local potentials and positive energy bound states , Libyan Journal of Science,13,69-70. [3] Husain D., Suhrabuddin S., Awin A. M.(1985) Scattering at the positive energy bound states , Australian Journal of Physics, 38,539-545. [4] Awin A. M.(1990) Positive energy bound states at higher order partial waves, Fizika , 22,3,513-519. [5] Awin A. M.(1991) Positive energy bound states, Fortschritte der Physik,89,2,131-158. [6] Abramowitz M. ,Stegun I.A.(1972) Handbook of Mathematical Functions,8th Edition, Dover Publications Incorporation, New York,447-466. [7] Muang K.M. ,Deutchman P.A. ,Royalty W.(1989) Integrals involving the three-parameter Fermi function ,Canadian Journal of Physics,67(2-3),95-99. [8] Fayyad H.M., Rihan T.H. , Awin A.M.(1996) An inversion solution to heavy-ion optical model potential at intermediate energies, Physical Review C `,53,5,2334-2340. [9] Karatsuba E.(2001) Fast computation of some special integrals of mathematical physics ,pp 28-40 DOI 10.1007/978-1-4787-6454-0_3 https://www.researchgate.net/publication/257366768

Copyright

Copyright © 2024 Efaf A. Aljatlawi, Ali M. Awin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.