# 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 '12 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 '12 at 11:46

Of course, Jon! Thanks! Unfortunately, the symbol $J_0()$ is not used in "Théorie" ... – Papiro May 19 '12 at 21:58
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 '12 at 14:53