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

426

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

and so we have the formula
(8) [cos -Ivrv. J, (ax) + sin vr. H (ax)] x =+ = - {tv (a)-L (ak)},
where a > 0, R (k) > 0 and - I < R (v) < 2. The change in the order of the
integrations presents no great theoretical difficulties.
A somewhat similar integral is
f x^K, (ax) dx
x' + k2
which converges if R (v) > - 1 and R (a) > 0.
If we choose k so that R (k) > 0, we have, by ~ 616 (1),
xv:K^ (ax)dx r(v+ ) fv f: (2a)vcosxu.dudx
Jo 2 + k2 F (1) Jo Jo (x2 + k2) (t2 + a2)v+
7 r (v + I) f (2a)v e-uk dm
-2k r() J (U2 + a2)v+i
= 4rcos H_v (ak) - Y_, (ak)},
when we use ~ 10'41 (3). Hence, when R (v) > -,
(9) xI = cvKv (ax) dx 7r_ (I)}
x2 + k2 4 cos vr H,(a)- Y, (c),
and therefore, when R (v) < 2,
(10) f:a - 4/c + =. cos- {H (ak) - Yv (ak)}.
Jo + kx 2 xv 4k"+l'cOS V77- *v/ v\ ^
These formulae (when v = 0) are due to Nicholson, and the last has also
been given by Heaviside.
f*1 Jv(ax) dx
The integral Jv (a d09~ x2 + k2 xv
has been investigated by Gegenbauer*. To evaluate it, we suppose that
R (v) > - - and that a and R (k) are both positive; we then have
f J (ax) dx 2 (a) rf cos (ax cos )
( a)v e-k cos o sin2 OdO,
r ( + )() r Jo 'k
and so
(11) J, (ax) dx r [I (ak)- L (ak).
x2 + k2 xv 2kv+i
* Wiener Sitzungsberichte, LXXII. (2), (1876), p. 349. Gegenbauer's result is incorrect because
he omitted to insert the term - Lv (ak); and consequently the results which he deduced from his
formula are also incorrect. A similar error was made by Basset, Proc. Camb. Phil. Soc. vi. (1889),
p. 11. The correct result was given by Gubler, Zurich Vierteljahrsschrift, XLVII. (1902),
pp. 422-424.