TY  - 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 . .