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## Fourier and Bessel

Oliver Heaviside, on page 387 of Electrical Papers, Vol. I, Macmillan and Co., 1892, available here, writes

$$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots = J_0(nr)$$

and

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.

Heaviside gives no reference.

I have two questions:

1. Are there any references in the Fourier work about the symbol $J_0()$ ?
2. What does the letter $J$ stand for?

Any references would be appreciated.

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According to Cajori's history of mathematical notation, Bessel used the letter I and Hansen changed it to J. If this is true, then it almost certainly has nothing to do with the J in Joseph Fourier. – Henry Cohn May 18 2012 at 11:44
Incidentally, Dutka's article "On the early history of Bessel functions" (dx.doi.org/10.1007/BF00376544) does not seem to discuss the naming issue, but goes back even further than Fourier (to the Bernoullis and Euler). – Henry Cohn May 18 2012 at 11:46

There is a fundamental reference using Bessel functions in Fourier's works. This is "ThÃ©orie analytique de la chaleur" 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).

The modern nomenclature was invented by Bessel himself on 1824, just two years after Fourier's work. This is proved in F. Bessel, "Untersuchung des Theils der planetarischen StÃ¶rungen", Berlin Abhandlungen (1824). Here the functions I and J get their names.

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Of course, Jon! Thanks! Unfortunately, the symbol $J_0()$ is not used in "Théorie" ... – PaPiro May 19 2012 at 21:58
You are right but I will fix it. – Jon May 20 2012 at 9:20
+1 on the "of course" (of course). – Emilio Pisanty May 20 2012 at 12:29
Actually, the 1824 Bessel paper does not quite use the modern notation. Instead, it uses $I$ where we use $J$. For example, the table on page 46 shows $J_0(k)$ and $J_1(k)$ in modern notation. – Henry Cohn May 20 2012 at 14:53
@Henry: Nice comment. But the first use is surely there. – Jon May 20 2012 at 15:52