Fourier and Bessel

2012-05-18T11:19:11Z

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:

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

Any references would be appreciated.

Answer by Jon for Fourier and Bessel

2012-05-19T12:25:41Z

There is a fundamental reference using Bessel functions in Fourier's works. This is "Théorie 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 Störungen", Berlin Abhandlungen (1824). Here the functions I and J get their names.

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Fourier and Bessel

Answer by Jon for Fourier and Bessel