E704 -- Disquisitio ulterior super seriebus secundum multipla cuiusdam anguli progredientibus

(English Translation of Title)


Euler starts with a cosine series f(f) = a0 + a1cos f + a2cos(2f) + a3cos(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)) = a0 + an + a2n + ..., 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 a0 = (1/p)∫0p f(f) df and an = (2/p)∫0p f(f) cos(nf) df.

According to the records, it was presented to the St. Petersburg Academy on May 29, 1777.

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