The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind
Апстракт
Recently, the Fourier series expansions of the Legendre incomplete elliptic integrals F(phi, k) and E(phi, k) of the first and second kind in terms of the amplitude phi were investigated and found in a series of papers. The expansions were derived in several ways, for instance, by using a hypergeometric series approach, and have coefficients involving either the hypergeometric function or the associated Legendre functions of the second kind. In this paper, it is shown that the Fourier series expansions of F(phi, k) and E(phi, k) can be obtained without any difficulty by applying the usual and more familiar Fourier-series technique. Moreover, as an interesting consequence of this approach, both the recently found expansions and the new expansions with coefficients which are solely linear combinations of the complete elliptic integrals of the first and second kind, K(k) and E(k), are obtained in a unified manner. Furthermore, unlike the previously known, the newly established results mak...e it possible to easily compute the Fourier coefficients of F(phi, k) and E(phi, k) analytically.
Кључне речи:
Fourier series / incomplete elliptic integral of the first kind / incomplete elliptic integral of the second kindИзвор:
Integral Transforms and Special Functions, 2010, 21, 3, 235-242Финансирање / пројекти:
- Ministry of Science and Technological Development of the Republic of Serbia [142025, 144004]
DOI: 10.1080/10652460903178552
ISSN: 1065-2469
WoS: 000273668200006
Scopus: 2-s2.0-77950965781
Колекције
Институција/група
VinčaTY - JOUR AU - Cvijović, Đurđe PY - 2010 UR - https://vinar.vin.bg.ac.rs/handle/123456789/3876 AB - Recently, the Fourier series expansions of the Legendre incomplete elliptic integrals F(phi, k) and E(phi, k) of the first and second kind in terms of the amplitude phi were investigated and found in a series of papers. The expansions were derived in several ways, for instance, by using a hypergeometric series approach, and have coefficients involving either the hypergeometric function or the associated Legendre functions of the second kind. In this paper, it is shown that the Fourier series expansions of F(phi, k) and E(phi, k) can be obtained without any difficulty by applying the usual and more familiar Fourier-series technique. Moreover, as an interesting consequence of this approach, both the recently found expansions and the new expansions with coefficients which are solely linear combinations of the complete elliptic integrals of the first and second kind, K(k) and E(k), are obtained in a unified manner. Furthermore, unlike the previously known, the newly established results make it possible to easily compute the Fourier coefficients of F(phi, k) and E(phi, k) analytically. T2 - Integral Transforms and Special Functions T1 - The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind VL - 21 IS - 3 SP - 235 EP - 242 DO - 10.1080/10652460903178552 ER -
@article{ author = "Cvijović, Đurđe", year = "2010", abstract = "Recently, the Fourier series expansions of the Legendre incomplete elliptic integrals F(phi, k) and E(phi, k) of the first and second kind in terms of the amplitude phi were investigated and found in a series of papers. The expansions were derived in several ways, for instance, by using a hypergeometric series approach, and have coefficients involving either the hypergeometric function or the associated Legendre functions of the second kind. In this paper, it is shown that the Fourier series expansions of F(phi, k) and E(phi, k) can be obtained without any difficulty by applying the usual and more familiar Fourier-series technique. Moreover, as an interesting consequence of this approach, both the recently found expansions and the new expansions with coefficients which are solely linear combinations of the complete elliptic integrals of the first and second kind, K(k) and E(k), are obtained in a unified manner. Furthermore, unlike the previously known, the newly established results make it possible to easily compute the Fourier coefficients of F(phi, k) and E(phi, k) analytically.", journal = "Integral Transforms and Special Functions", title = "The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind", volume = "21", number = "3", pages = "235-242", doi = "10.1080/10652460903178552" }
Cvijović, Đ.. (2010). The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind. in Integral Transforms and Special Functions, 21(3), 235-242. https://doi.org/10.1080/10652460903178552
Cvijović Đ. The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind. in Integral Transforms and Special Functions. 2010;21(3):235-242. doi:10.1080/10652460903178552 .
Cvijović, Đurđe, "The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind" in Integral Transforms and Special Functions, 21, no. 3 (2010):235-242, https://doi.org/10.1080/10652460903178552 . .