E704  Disquisitio ulterior super seriebus secundum multipla cuiusdam anguli progredientibus
(English Translation of Title)
Summary:
Euler starts with a cosine series f(f) = a_{0} + a_{1}cos f + a_{2}cos(2f) + a_{3}cos(3f) and notes that if n > 0 is an integer, then (1/n)((1/2)f(0) + f(w) + f(2w) + ... + f((n  1)w) + (1/2)f(p)) = a_{0} + a_{n} + a_{2n} + ..., where w = p/n. He gets a similar formula for values shifted by n by a trigonometric addition formula. He then gives the Fourier theorem that a_{0} = (1/p)∫_{0}^{p} f(f) df and a_{n} = (2/p)∫_{0}^{p} f(f) cos(nf) df.
According to the records, it was presented to the
St. Petersburg Academy on May 29, 1777.
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 11, 1798, pp. 114132

Opera Omnia: Series 1, Volume 16, pp. 333  353
Documents Available:
 Original Publication: E704
 German translation (Alexander Aycock and Artur Diener): E704
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E704 include:
 Dhombres J., “Some aspects of the history of functionalequations linked to the evolution of the function concept.” Archive for History of Exact Sciences, 36 (2), pp. 91181 (1986).
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