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

3-21, 3-3]

BESSEL FUNCTIONS

47

with numerous other formulae of like character. These results seem to be of
no great importance, and consequently we merely refer the reader to the
memoirs in which they were published.
In the special case in which v = 0, Bessel's equation becomes
2 d2y dy
z d + z + (z2 + 2) y = 0;
solutions of this equation in the form of series were given by Boole* many
years ago.
3'3. Lomrnel's expression of J, (z) by an integral of Poisson's type.
We shall now shew that, when R () > - -, then
(1) Jv (z) = (2)( 1 cos (z cos 0) sin2 0dO.
It was proved by Poissont that, when 2v is a positive integer (zero included), the expression on the right is a solution of Bessel's equation; and
this expression was adopted by Lommel+ as the definition of Jv (z) for positive
values of v + 2.
Lommel subsequently proved that the function, so defined, is a solution of Bessel's
generalised equation and that it satisfies the recurrence formulae of ~ 3'2; and he then
defined J, (z) for values of v in the intervals (-, - 3), (- ', - ), (-_, - _),... by successive applications of ~ 3'2 (1).
To deduce (1) fromn the definition of J, (z) adopted in this work, we transform the general term of the series for J, (z) in the following manner:
(-)m (1z)+v2m (_)m (z)v. m Pr (v + - ) P (rm + L)
m! r(v + m+l) (v+ () (2m)! r( + m+l)
=r m (z2+ r )( )ot-)vt (1 -t)m- dt,
provided that R (v)> -.
Now when R (v) >-, the series
oo (_)m 2M1
=1 (2rn)!
converges uniformly with respect to t throughout the interval (0, 1), and so it
may be integrated term-by-term; on adding to the result the term for which
* Phil. Tramas. of the Royal Soc. 1844, p. 239. See also a question set in the Mathematical
Tripos, 1894.
t Journal de l'Ecole R. Polytechnique, xII. (cahier 19), (1823), pp. 300 et seq., 340 et seq.
Strictly speaking, Poisson shewed that, when 2v is an odd integer, the expression on the right
multiplied by,/z is a solution of the equation derived from Bessel's equation by the appropriate
change of dependent variable.
+ Studien ilber die Bessel'schen Functionen (Leipzig, 1868), pp. 1 et seq.