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The $n$th Bessel function of the first kind is usually denoted $J_n(x)$.

Where did the use of the letter $J$ to indicate the Bessel function come from?

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  • $\begingroup$ Look at the first paragraph in this mathworld entry for Bessel functions. It appears that you should consult Cajori's book from the references given there. $\endgroup$ Apr 15, 2011 at 22:14
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    $\begingroup$ Well, it's the notation used by Bessel himself, see here. As to why Bessel used this notation, a cursory glance seems to indicate that he was just following the alphabet. $\endgroup$
    – t.b.
    Apr 16, 2011 at 7:32
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    $\begingroup$ On the other hand, for the "second kind" Bessel functions, $N_\nu(z)$ was used, for "Neumann"; how they became denoted $Y_\nu(z)$ I have forgotten. I'll check my handbooks and report back. $\endgroup$ Apr 16, 2011 at 16:58
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    $\begingroup$ @t.b., can you add your comment as an answer? $\endgroup$ Mar 21, 2012 at 10:47

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" Bessel defined the function now known by his name by the following definite integral: $$\int \cos (h\epsilon-k \sin \epsilon) d\epsilon=2\pi I_k^h$$ where $h$ is an integer and the limits of integration are $0$ and $2\pi$. His $I_k^h$ is the same as the modern $J_h(k)$, or rather $J_n$(x).

O. Schlomilch following P. A. Hansen explained the notation $J_{\lambda±n}$ where $\lambda$ signifies the argument and $\pm$ the index of the function. Schlomilch usually omits the argument. Watson points out that Hansen and Schlomilch express by $J_{\lambda,n}$ what now is expressed by $J_n(2\lambda)$. Schlafly marked it $^nJ(x)$. Todhunter uses the sign $J_n(x)$. $J_n(x)$ is known as the "Bessel function of the first kind of order $n$," while $Y^n(x)$, an allied function introduced in 1867 by Karl Neumann is sometimes called "Newmann's Bessel function of the second kind of order $n$." It is sometimes marked $Y_n(x)$.

Watson says: "Functions of the types $J \pm (n+1/2)(z)$ occur with such frequency in various branches of Mathematical Physics that various writers have found it desirable to denote them by a special functional symbol. Unfortunately no common notation has been agreed upon and none of the many existing notations can be said to predominate over the others. He proceeds to give a summary of the various notations in his Theory of Bessel Functions, pages 789, 790, Watson gives a list of 183 symbols used by him as special signs pertaining to that subject. "

This, in outermost quotes is a paragraph taken from Cajori's Book "A history of Mathematical Symbols" , clause 664, pg 279, vol-II explaining the symbol chosen for Bessel Function of the first kind. Hope it helps. Cajori, F., A history of mathematical notations. Vol. II: Notations mainly in higher mathematics., XVIII + 367 p., 20 fig. Chicago, The Open Court Publishing Company (1929). ZBL55.0002.02.f

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    $\begingroup$ Thanks for useful answer! $\endgroup$
    – Black Mild
    Apr 10, 2022 at 19:58

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