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

16-41, 16-5]

LOMMEL S FUNCTIONS

537

In like manner, if
(3) f(z)= anzV+'J.+n Z),?n=0
then
rF(2v+2m+l) 1+2 V
- 2v+m r (v + m + l) an' = (2
[NOTE. If ea sin bz = E anJ, (z),
n=l
then (G2bz (1 + 2)
n=l (1 - 2az - z22 + 4b2z2
This result, which is immediately deducible from Cailler's theory, was set as a problem
in the Mathematical Tripos, 1896.]
16'5. Lommel's functions of two variables.
Two functions, which are of considerable importance in the theory of
Diffraction and which are defined by simple series of Neumann's type, have
been discussed exhaustively by Lommel* in his great memoirs on Diffraction
at a Circular Aperture and Diffraction at a Straight Edge.
The functions of integral order n, denoted by the symbols Un (w, z) and
Vn (w, z), are defined by the equations
() /\ Wf+2m
(1) Un (w. Z)=, (_)m Jn+m (Z)
(2) Yn (w. z)= E ()m -) J-n-2. (Z)m
m=O \
It is easy to see from ~ 2'22 (3) that
0o /W \ +2n
(3) Un (w, Z)- V_1+2 (w, Z) = (-) Jn+m (z)
m= -mo \/
/W l z 2 nnr7
=cos 2+ - 2)(4) Un+, (w, z) - V_-,+ (w, z)= sin ( + 2 - 2 )
The last equation may be derived from the preceding equation by replacing
n by n+ 1.
There is no difficulty in extending (1) to define functions of non-integral
order; for unrestricted values of v we write
00 w\ v+2m
(5) Uv w,z)= E (-)" I- J+2m (z)m=0 z
* Abh. der math. phys. Classe der k. b. Akad. der Wiss. (Miinchen), xv. (1886), pp. 229-328,
529-664. The first memoir deals with functions of integral order; and the definition of Vz (w, z)
in it differs from that adopted subsequently by the factor (- 1)". Much of Lommel's work is reproduced by J. Walker, The Analytical Theory of Light (Cambridge, 1904). The occurrence of Lommel's
functions in a different physical problem has been noticed by Pocklington, Nature, LXXI. (1905),
pp. 607-608.