# How did Fourier determine the coefficients of Fourier series?

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.

• 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. Feb 26, 2021 at 23:48
• 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. Feb 27, 2021 at 0:02
• 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. Feb 27, 2021 at 2:24
• Thank you for your effort, don't know how to express my gratitude to your help. Thanks! Feb 27, 2021 at 3:23
• @Alexandre Eremenko: Third, I very seriously doubt that Hawking did the translation. Feb 27, 2021 at 18:48