A treatise on the theory of Bessel functions, by G. N. Watson.

84 THEORY OF BESSEL FUNCTIONS [CHAP. III
Some writers*, following Heine t who called J,(z) a Fourier-Bessel function, call J,(z)
a Fourier function.
Although Bessel coefficients of any order were used long before the time of Bessel
(cf. ~~ 1'3, 1'4), it seems desirable to associate Bessel's name with them, not only because
it has become generally customary to do so, but also because of the great advance made by
Bessel on the work of his predecessors in the invention of a simple and compact notation
for the functions.
Bessel's name was associated with the functions by Jacobi, Journal fiir Math. xv.
(1836), p. 13 [Ges. Math. Werke, vi. (1891), p. 101]. "Transcendentium Iki naturam variosque usus in determinandis integralibus definitis exposuit ill. Bessel in commentatione
celeberrima."
A more recent controversy on the name to be applied to the functions is to be found in
a series of letters in Ntature, LX. (1899), pp. 101, 149, 174; LXXXI. (1909), p. 68.
* E.g. Nicolas, Ann. Sci. de 'Ecole norm. sap. (2) xi. (1882), supplement.
t Journal fir Math. LXIX. (1868), p. 128. Heine also seems to be responsible for the term
cylinder function.