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

660

THEORY OF BESSEL FUNCTIONS

[CHAP. XX

Bourget's Tables have been reprinted so frequently that their authorship has been
overlooked by the writers of the articles on Bessel Functions in the Encyclopddie der
Maath. Wiss. and the Encyclopedie des Sci. Math.
The first five zeros of J, (x) and J2 (x) were given to six places of decimals
by Lommel, Zeitschrift fiir Math. und. Phys. xv. (1870), p. 167 and Miinchener
Abhandlungen, xv. (1886), p. 315.
The first ten zeros of J (x) were computed to ten places of decimals by
Meissel, Berliner Abhandlungen, 1888.
The first fifty zeros (and their logarithms) of J, (x) were given to ten places
of decimals by Willson and Peirce, Bulletin American Math. Soc. III. (1897),
pp. 153-155; they also gave the values of J,(x) and log I J (x) I at these
zeros to eight and seven places of decimals respectively.
The first fifty zeros of J, (x) and the corresponding values of J (x) were
computed to sixteen places of decimals by Meissel*, Kiel Programm, 1890;
this Table is reprinted by Gray and Mathews in their Treatise, p. 280.
Tables of roots of the equation
Jn (X) Yn (kx) - J, (k) Y, () = 0
have been constructed by Kalahne, Zeitschrift fiir Math. und Phys. LIV. (1907),
pp. 55-86; the values taken for k are 1'2, 1'5 and 2'0, while n is given the
values 0, 2, 1, 2, 2, 2.
Dinnik in his Tables of functions of fractional order mentions the values
of a few of the zeros of each function, while Airey, Phil. Mag. (6) XLI. (1921),
pp. 200-205, has computed the value of the smallest zero of Jv (x) for small
fractional values of v by Euler's method.
Rayleigh, Proc. London Math. Soc. x. (1878), pp. 6-7 [Scientific Papers,
I. (1899), pp. 363-364], has calculated that
(I - X2) XI (x)/Io (x)
has a maximum when x2 = 0'4858.
Airey, Archiv der Math. und Phys. (3) xx. (1913), p. 291, has computed
the first ten zeros of 3xJo (x)- 2J, (x) and of 2x'Jo (x) - J (x) to four places
of decimals.
In his memoirs on Diffraction, Miinchener Abhandlungen, xv. (1886),
Lommel has published tables connected with his functions of two variables,
but these tables are so numerous that a detailed account of them will not be
given here. His Table of Fresnel's integrals (p. 648) to six places of decimals
from x= 0 to x = 50'0 with interval 0'5 (with auxiliary tables for purposes
of interpolation) must, however, be mentioned, and with it his Table of the
first sixteen maxima and minima of these integrals.
* Jahrbuch iiber die Fortschritte der Math. 1890, p. 521. In consequence of the inaccessibility
of Meissel's table, the zeros of J1 (x) were recomputed (to ten places of decimals) for insertion in
Table VII, p. 748.