The values of
for x=o(o.oi)i.00, n= 1(07 are given by Glaisher in Brit. Assoc. Rep. (5879). They are reproduced by Dale and Jahnke-Emde.
0) for n=i(i)7 and
are given to 4 places by Perry, Proc. Phys. Soc. (1892) and Phil. Mag. (1891) ; also in Jahnke-Emde. Extensive tables have been pub lished by Tallquist at Helsingfors, 1908.
The work of Jahnke and Emde referred to above contains a list of all tables before 1909 connected with Bessel Functions, and also some 4-place tables. The majority of tables of these functions are scattered through the journals of mathematical and scientific societies. Hansen's extension of Bessel's tables is reproduced by Schlomileh in Zeitschr. fur Math. (1868) and by Lommel in Studien caber die Bessel'schen Func tionen (1868). This is a 7-place table of
(x) and
for x=o to 20 at intervals of o.oi. These functions to 12 decimals are given by Gray and Mathews as a reprint of the table of Meis sel, Abh. d. Berlin Akademie (1888). In 1889 the Commit tee of the British Association for the Calculation of Tables de cided to commence work on the tables of Bessel Functions and the first extensive table appeared in 1893, giving
to 9 places with first differences for x=o.000(o.00i)5.ioo;
followed in 1896. Further tables appeared in the Reports of this Committee in 1907, 1909 and 1911 [also Neumann Functions
and
1912 (ber x and bei x), 1913 and later. In Proc. Roy.
(1899) Aldis, "On the numerical computation of the func tions
and .4(x)" gives a number of tables to 21 places for x=o.o(o.i)6.o.
The Committee of the British Association (Mathematical Tables) commenced work on the elliptic functions in 1911 and tables appeared in the Reports of 1911-12-13-19. In the Report of 1924, tables of the Lommel Weber and Bessel-Clifford functions are given.
Legendre, Traite des fonctions elliptiques (1826), gives a num ber of tables of the elliptic functions of the first and second kind and 4-place tables appear in the Jahnke-Emde collection.
from 0=o° to 9o° has been tabulated by Glaisher in Month. Not. R. A. Soc. (1877) for every degree to io places; Jacobi, Crelle's Journal, XXVI., for every tenth of a
degree to 5 places; Bertrand, Calcul Integral (187o), for every 5 minutes to 5 places and Meissel, Sammlung mathematischer Tafeln (Iserlohn, 186o) for every minute to 8 places.
Tables of integrals such as Si
Cix= f x oc
x 0 x _d Ei x = f -e x
Li x= f have also been published. The 17) X f og Jahnke-Emde collection contains tables of the first three integrals to 4 places; Glaisher Phil. Trans. (1870), P.
gives Si x, Ci x, Ei x to 18 places for x=o.00(o.oi)i.o, and to II places for x = i (o.1) 5(1)15(5)20. Ei x is also given by Bretschneider in Grunert's Archiv III., p. 33 to 20 places for x= r (Om and for x= '0(r)20 by Gram in Publications of the Copenhagen Academy
The latter extends Glaisher's table in some places, giving Ei x for x=5.o(o.2)20.0 to 8, 9 or io places. Li x has been published by Glaisher in his Factor Tables (1883) to the nearest integer for 0(50,000)9,000,00o. Tables of the integrals are of limited extent.
In the Brit. Assoc. Report (1926) C(x) and S(x) were tabulated to 6 places for x= o.o(o.i) 20.0. Lommel, Abh. Munch. Akademie (188o), gives the values for z=o(I)5o, where the integral
is written in the form
I rz sin
A 4-place table appears also in Jahnke-Emde.
z Pearson Integral.—The integral F (r ,n) = eTn f sinrx e" dx 0 which is required in certain statistical work has been tabulated in the form logF(r,4) where n=r tang), in Brit. Assoc. Reports (1896) and (1899), for successive integral values of r to 5o, to 7 places. Tables of associated integrals appear in Tables for Statisticians.
References.—Glaisher, "Report on Mathematical Tables," Brit. Assoc. Report (1873) ; De Morgan, "Mathematical Tables," English Cyclo paedia (1861), vol. vii.; Mehnke, "Numerisches Rechnen," Encyk. der Math. Wiss. (Leipzig, 1900-04) , vol. i., pt. ii., pp.
Horsburgh, Modern Instruments and Methods of Calculation (Edinburgh, 1915), PP.
; Henderson, Bibliotheca Tabularum Mathematicarum, being a descriptive catalogue of mathematical tables, Part I. Logarith mic Tables (Cambridge University Press, 1926) (in progress).
H.)