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

80 THEORY OF BESSEL FUNCTIONS [CHAP. III
These results are due to Basset. We also have
(10) In+2 (Z) V(2T z) H r-or! r(n-)!(2z)i
+ (- l e I (n+r)! ]
(11) '-(~+) (z) = [ r (n
=V0(27rz) L - r+ (n - r)! (2z)4
(n + r)!
(12)! (o - r)! (2z)rj
/ n 4
(1) 1 [ T / \ (-) (,+ /. )!
(13) 1i, (z)= ( e e
(~) mi (n $ m)! ~lv ~ 0 7H - rl 2, r
(14) o (z) = - log ()). I (z) + z Z m (r + 1),
(15) n ()m (n — ) - -!
co /(1 2n+2m
+ (-)+t m (+)! {log (.z)-t ~ (i + l) - (n~ + + 1)},
(16) Ko (z) = - ez~osO {log (2z sin2 0) + 7} dO,
7r.' o
(17) I (emri) = e-mvi Iv (z),
(18) K, (zen'i)= e-"="i K, (z) - ri n r (z),
(19) m[ {I (z), K; (z)} =- G /Z,
(20) I, (z) KC+ (z) + I1+ (z) 1, (z)= 1/.
The integral involved in (16) has been discussed by Stokes (cf. ~ 3'572).
The integrals involved in (9) and the series in (14) were discussed by Riemann in his
memoir "Zur Theorie der Nobili'schen Farbenringe," Ann. der Physik und Chemie, (2) xcv.
(1855), pp. 130-139, in the special case in which v=0; he also discussed the ascending
power series for J0 (z).
The recurrence formulae have been given by Basset, Proc. Canzb. Phil. Soc. vI. (1889),
pp. 2-19; by Macdonald, Proc. London Math. Soc. xxIx. (1899), pp. 110-115; and by
Aichi, Proc. Phys. Math. Soc. of Japan, (3) II. (1920), pp. 8-19.
Functions of this type whose order is half an odd integer, as in equations (10) and (12),
were used by Hertz in his Berlin Dissertation, 1880 [G(es. Terke, I. (1895), pp. 77-91];
and he added yet another notation to those described in ~ 3'41.