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

10-61,10-62]

ASSOCIATED FUNCTIONS

343

With the aid of these formulae combined with the corresponding formulae
for J, (z), Y, (z) and Sn (z), we deduce from ~ 10'6 (1) that
(6) Un,_ (z) + Un (z) = (2n/z) U (z) - (2/z) Jn (z),
(7) U._-1 (Z) - Un+ (z) = 2 U () - (2/z) J, (z),
(8) ( S + n) Un (z) = z u,_ (z) + 2 J (z),
(9) ( - n) Un (z) = - z Un+ (z), [cf. ~~ 3-58 (1), 3-58 (2)]
(10) V,, U (z) =- 2zJ,+ (z).
The reader may verify these directly from the definition, ~ 10'6 (3).
It is convenient to define the function Tn (z), of negative order, by the
equivalent of ~ 10'6 (4). If we replace 0 by r - 0 in the integral we find that
2 ftr
T — (z)= -2 (2 7T-0) sin (z sin 0 + n0) d0
-= - -o0) si (z - sn in 0 - n + n7r) dO,
77 Jo
and so
(11) )7-1 (Z) = (_-)n+ T (Z).
We now define U_, (z) by supposing ~ 10'6 (1) to hold for all values of n;
it is then found that
(12) _n (z) (-) {Un (2) - T (z) + Sn (z)}.
10-62. Series for Tn (z) and U,, (z).
We shall now shew how to derive the expansion
X, 1
(1) ]i(z) t 1[Jn (^-J (z)}
(Z) {Jn~2 M (Z) - Jn2m2 (Z)}
m=l 971
from ~ 10'6 (4). The method which we shall use is to substitute
l - = sin 2m0
27r - = E
in the integral for T (z), and then integrate term-by-term. This procedure
needs justification, since the Fourier series does not converge uniformly near
0 = 0 and 0 = rr, and, in fact, the equation just quoted is untrue for these two
values of 0.
To justify the process', let 8 and e be arbitrarily small positive numbers. Since the
series converges uniformly when 8 < 0 r - 8, we can find an integer mo such that
M sin 2oO
m=l -n <
* The analysis immediately following is due to D. Jackson, Palermo Rendiconti, xxxii. (1911),
pp. 257-262. The value of the constant A is 1 8519....