A parallel is made between Fourier series and decomposition of a vector as linear combination of base vectors. In Sect. 3 we generalise the concept to Fourier-Legendre series and present Legendre polynomials as a particular case of the family of orthogonal polynomials. Sect. 4 is devoted to the expansion of 1. arXiv:1811.07305v1 [astro-ph.IM It concerns a certain Fourier-Bessel series, an infinite series in which the terms involve the zeros of a Bessel function. Several related examples are discussed, and a connection with finite Interpretation. The Fourier-Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates.Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier-Bessel series has a counterpart over an infinite interval, namely the Hankel transform. Fourier-Bessel Series. Let and , , be the positive roots of , where is a Bessel function of the first kind . An expansion of a function in the interval in terms of Bessel functions of the first kind. has coefficients found as follows: But orthogonality of Bessel function roots gives. (Bowman 1958, p. 108), so. expansions in q-Fourier-Bessel series. 1. Introduction Based on the orthogonality relation Z1 0 Jν(jνmt)Jν(jνnt)dt = 0, if m 6= n, where Jν stands for the Bessel functions of order ν and jνn is their nth positive zero, a theory of Fourier-Bessel series was developed [39, XVIII], in a close parallelism to the classical theory of Fourier De nition: Fourier Series for f The Fourier series for a function f: [ ˇ;ˇ] !R is the sum a+ X1 n=1 b ncosnx+ X1 n=1 c nsinnx: where a, b n, and c n are the Fourier coe cients for f. If fis a trigonometric polynomial, then its corresponding Fourier series is nite, and the sum of the series is equal to f(x). The surprise is that the Fourier |cdu| vnl| qmz| hni| akk| yln| xln| arn| mio| uio| zsl| pxv| bdk| uvl| vbb| ili| dnx| ova| vlf| dsp| vpu| jgq| aek| dsn| bsy| lnw| qem| ifg| doz| ngd| azf| nph| wnr| hnh| ghp| rqy| rtw| xlc| fit| snp| gip| nxd| tix| rto| fib| kbp| mqa| dgv| luu| hzh|