Open AccessFourier's series and analytic functions
Publication year - 1922
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1922.0031
Subject(s) - fourier series , complex plane , mathematics , series (stratigraphy) , lebesgue integration , interval (graph theory) , analytic function , constant (computer programming) , function (biology) , fourier transform , locally integrable function , mathematical analysis , integrable system , pure mathematics , combinatorics , computer science , paleontology , evolutionary biology , biology , programming language
1.1. In the theorems which follow we are concerned with functionsf (x ) real for realx and integrable in the sense of Lebesgue. We do not, however, remain in the field of the real variable, for we suppose, in 4et seq ., thatf (x ), or a function associated withf (x ), is analytic, or, at any rate, harmonic, in a region of the complex plane associated with the particular real value ofx considered. The Fourier’s series considered are those associated with the interval (0, 2π). Ifa is a point of the interval, we writeϕ(u) = ½ {f (a +u ) +f (a -u ) - 2s } (0 <a < 2π), (1.11)ϕ(u) = ½ {f (u ) +f (2π -u ) - 2s } (a = 0,a = 2π), (1.12) wheres is a constant.