# How did Bessel functions come to be denoted by $J_n$?

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?

• 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. Apr 15, 2011 at 22:14
• 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.
– t.b.
Apr 16, 2011 at 7:32
• 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. Apr 16, 2011 at 16:58
" 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. "