I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the integers". I fail to find a reference for how Fourier determine the coefficients of the Fourier series. Fourier, in my opinion, should be ranked as one the greatest mathematicians in the 19th century for he laid a great foundation on the development of trigonometric series, an essential area of modern mathematics.

Can you summarize his findings and point out which part of his work contains his derivation? I am greatly interested in how he was able to determine the coefficients of Fourier series, a task that Daniel Bernoulli failed to achieve.

I hope my shallow knowledge of mathematics do not offend and repel you from this question.

  • $\begingroup$ First of all, Fourier's book, Analytic theory of heat is freely available online in English translation: www3.nd.edu/~powers/ame.20231/fourier1878.pdf (and also in the original), so you can easily look yourself. Second, he determined Fourier coefficients in exactly the same way that modern undergraduate students do. $\endgroup$ Feb 26, 2021 at 23:48
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    $\begingroup$ But which part of the book is relevant? Can you give me a reference? It is a thick book. If you know which chapter I can find, that would be highly valuable. $\endgroup$ Feb 27, 2021 at 0:02
  • $\begingroup$ The book has a table of contents, by the way. Discussion of Fourier series is in Chap III, sections IV - VI. The formula for Fourier coefficients is written for the first time on p 185, formula D. $\endgroup$ Feb 27, 2021 at 2:24
  • $\begingroup$ Thank you for your effort, don't know how to express my gratitude to your help. Thanks! $\endgroup$ Feb 27, 2021 at 3:23
  • $\begingroup$ @Alexandre Eremenko: Third, I very seriously doubt that Hawking did the translation. $\endgroup$ Feb 27, 2021 at 18:48


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