I located Joseph Fourier's book, Analytical Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?

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    $\begingroup$ If you intend to pursue historical issues, then you need to be a bit more nuanced than saying something like Fourier's 1878 book. $\endgroup$ Commented May 2, 2020 at 18:53
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    $\begingroup$ Probably the place you want to look is Joseph Fourier 1768–1830 by Ivor Grattan-Guinness, which gives a very detailed analysis of Fourier's 1807 paper, later expanded in 1811 for a prize competition (but not published), and revised and expanded again as a book published in 1822. I believe the Fourier integral did not appear in the 1807 paper, but it did appear in the 1811 manuscript. But I don't have time now to track all this down more thoroughly, so if no one else says much more, then I recommend getting a copy of the book I cited. $\endgroup$ Commented May 2, 2020 at 19:09
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    $\begingroup$ Fourier transform is introduced in this book. $\endgroup$ Commented May 2, 2020 at 20:41
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    $\begingroup$ He wrote it as a combination of sine and cosine transforms. It's in the last chapter of the book, article 418ff. Perhaps, you should first read modern commentary, like Highlights in the History of the Fourier Transform, that explains it in modern notation. $\endgroup$
    – Conifold
    Commented May 2, 2020 at 21:07
  • $\begingroup$ Fourier elaborated the Fourier Series for the purpose of the theory of heat. It has been rigorously defined later on (all the 19th century). The Fourier transform is something quite different, elaborated by Wiener in the 1930s,... Truely speaking, there is a larger theory gathering Fourier series (for periodic functions) and Fourier transform. $\endgroup$ Commented Nov 19, 2020 at 12:34


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