Calculation of Sommerfeld Integrals in Dipole Radiation Problems
This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by...
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/298 |
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| author | Seil Sautbekov Merey Sautbekova Kuralay Baisalova Mustakhim Pshikov |
| author_facet | Seil Sautbekov Merey Sautbekova Kuralay Baisalova Mustakhim Pshikov |
| author_sort | Seil Sautbekov |
| collection | DOAJ |
| description | This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI). |
| first_indexed | 2024-03-08T10:41:56Z |
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| id | doaj.art-0b8e5507d1fd47d7a4d7fd648124c83d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:41:56Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-0b8e5507d1fd47d7a4d7fd648124c83d2024-01-26T17:33:10ZengMDPI AGMathematics2227-73902024-01-0112229810.3390/math12020298Calculation of Sommerfeld Integrals in Dipole Radiation ProblemsSeil Sautbekov0Merey Sautbekova1Kuralay Baisalova2Mustakhim Pshikov3Department of Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanDepartment of Mechanical Mathematics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanDepartment of Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanDepartment of Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanThis article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).https://www.mdpi.com/2227-7390/12/2/298Maxwell’s equationsconvolutionGreen’s functionelectromagnetic wave scatteringHertz dipoleSommerfeld integrals |
| spellingShingle | Seil Sautbekov Merey Sautbekova Kuralay Baisalova Mustakhim Pshikov Calculation of Sommerfeld Integrals in Dipole Radiation Problems Mathematics Maxwell’s equations convolution Green’s function electromagnetic wave scattering Hertz dipole Sommerfeld integrals |
| title | Calculation of Sommerfeld Integrals in Dipole Radiation Problems |
| title_full | Calculation of Sommerfeld Integrals in Dipole Radiation Problems |
| title_fullStr | Calculation of Sommerfeld Integrals in Dipole Radiation Problems |
| title_full_unstemmed | Calculation of Sommerfeld Integrals in Dipole Radiation Problems |
| title_short | Calculation of Sommerfeld Integrals in Dipole Radiation Problems |
| title_sort | calculation of sommerfeld integrals in dipole radiation problems |
| topic | Maxwell’s equations convolution Green’s function electromagnetic wave scattering Hertz dipole Sommerfeld integrals |
| url | https://www.mdpi.com/2227-7390/12/2/298 |
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