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

142

THEORY OF BESSEL FUNCTIONS

[CHAP. V

These may be proved by expressing the functions of the second kind as a
linear combination of functions of the first kind; by proceeding to the limit
when v tends to an integral value, we see that they hold for functions of
integral order.
By combining (11)-(14) with the corresponding results for functions of
the first kind, we see that we may substitute the symbol V for the symbol
Y throughout.
These last formulae were noted by Lommel, Studien, p. 87. Numerous generalisations
of them will be given in Chapter xi. It has been observed by Airey, Phil. Mcag. (6) xxxvi.
(1918), pp. 234-242, that they are of some use in calculations connected with zeros of
Bessel functions.
When we combine (5) and (13), and then replace \/(1 + k) by X, we find
that, when I 2- 1 1 < 1,
(_ C)\m ('2 ) I m(l,\m
(15) C () (Xz) = X"v ) m2 7 (Z)
m=O =v
and, in particular, when X is unrestricted,
( /-)Mm (\2 _ \m(MZ)m
(16) J (XZ) = X - ) ( )( - 2 Jv+m ().
mn =O 7/,
These two results are frequently described* as multiplication theorems for
Bessel functions.
It may be observed that the result of treating (14) in the same way as (8) is that
(when v is taken equal to an integer n)
(17) -(n-1)! (2/Z)=w E 2-,_(z).
m=O m.
An alternative proof of the multiplication formula has been given by B6hmer, Berliner
Sitzungsberichte, xiii. (1913), p. 35, with the aid of the methods of complex integration;
see also Nielsen, Math. Ann. LIx. (1904), p. 108, and (for numerous extensions of the
formulae) Wagner, Bern Mittheilungen, 1895, pp. 115-119; 1896, pp. 53-60.
[NOTE. A special case of formula (1), namely that in which v=l, was discovered by
Lommel seven years before the publication of his treatise; see Archiv der Math. xxxvII.
(1861), p. 356.
His method consisted in taking the integral
2- f cos (6r cos 9 + br sin 0) d$ dr
over the area of the circle 2 +r,2 = 1, and evaluating it by two different methods.
The result of integrating with respect to q is
21 fIlLsin (r cos + r sin )/(1-I 2) d
27r r-1 _J-wO-~)'v sin 8
1= cos ($r cos 0) sin {,(1 _- 2). *r sin } d
7r i r sin 9
1E (-)'" (r sin _)2m p I
=. =o (2)~ - ) j cos (^ cos 0). (1- 2)) d$
m r=0 (29n + 1)!
( - )' (~r sin 8)2m J +1 (r cos 0)
m=O m! (r cos )'+ 1 '
* See, e.g. Schafheitlin, Die Theorie der Besselschen Funktionen (Leipzig, 1908), p. 83.