Fourier and Bessel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:59:28Z http://mathoverflow.net/feeds/question/97298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97298/fourier-and-bessel Fourier and Bessel PaPiro 2012-05-18T11:19:11Z 2012-06-06T08:27:52Z <p>Oliver Heaviside, on page 387 of <em>Electrical Papers</em>, Vol. I, Macmillan and Co., 1892, available <a href="http://archive.org/stream/electricalpapers01heavuoft#page/386/mode/2up" rel="nofollow">here</a>, writes </p> <p>$$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots = J_0(nr)$$</p> <p>and </p> <p><strong>This function is usually denoted by $J_0(nr)$, and was first employed by Fourier. Whether he invented it or discovered it is a doubtful point; the question is raised whether mathematical truths lie within the human mind alone, or whether the infinite body of known and unknown mathematics could exist in a dead universe. But this is metaphysics, which is all vanity and vexation of spirit.</strong></p> <p>Heaviside gives no reference.</p> <p>I have two questions:</p> <ol> <li>Are there any references in the Fourier work about the symbol $J_0()$ ?</li> <li>What does the letter $J$ stand for?</li> </ol> <p>Any references would be appreciated.</p> http://mathoverflow.net/questions/97298/fourier-and-bessel/97389#97389 Answer by Jon for Fourier and Bessel Jon 2012-05-19T12:25:41Z 2012-05-20T10:20:45Z <p>There is a fundamental reference using Bessel functions in Fourier's works. This is <a href="http://books.google.it/books?id=TDQJAAAAIAAJ&amp;printsec=frontcover&amp;hl=it#v=onepage&amp;q&amp;f=false" rel="nofollow">"Théorie analytique de la chaleur"</a> firstly published on 1822. You will find this series firstly given in chapter VI pag. 370. This chapter is about the propagation of the heat in a cylinder ("of course" let me add).</p> <p>The modern nomenclature was invented by Bessel himself on 1824, just two years after Fourier's work. This is proved in <a href="http://edoc.bbaw.de/volltexte/2008/735/pdf/21hekEZuXxSQ.pdf" rel="nofollow">F. Bessel, "Untersuchung des Theils der planetarischen Störungen", Berlin Abhandlungen (1824)</a>. Here the functions I and J get their names.</p>