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

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DIFFERENTIAL EQUATIONS

131

and since such solutions are expressible as the product of a function of r and
a function of q they must be annihilated by each of the operators
a2 3 a 2 (4( + 1)
+- -- + -----
2 + n2~ +2
3.2 ~,\a4/~? ___
a- + (cot -- tan +) + 4 X (X + 1) - s n
+(c2 o 1 cos2 ^ sin2 (4
where X is a constant whose value depends on the particular solution under
consideration. The normal solutions so obtained are now easily verified to be
of the form
(kr)-1 J2+1 (kr) cos b sin,,
A, Y-+V -, O h + v + 1; +1; sin2 ei( + ^s+kn.
It is therefore suggested that
J,. (kr cos 4 cos )J) J, (kr sin 4 sin 4D)
is expressible in the form
E a, (kr)-1 J21 (kr) cost q sin,. 2F, (+ - X, /- + X + 1; v + 1; sin2),
h 2 2
where the summation extends over various values of X, and the coefficients aCt
depend on X and SI, but not on r or 4. By symmetry it is clear that
ak = b cos A)sinI.2F1 (- 2 -X, 2 +X + 1; v + l; sin 2 ),
where b6 is independent of <>.
It is not difficult to see that
X = 1 (4 + v)+,n, (n =, 0, 2,...)
and Bateman has proved that
b 2(-) (+ + 2n+1) (^ + n+ 1) P ( +n1)n )
II! P(i+ + ){I (V+ 1)}2
We shall not give Bateman's proof, which is based on the theory of linear
differential equations, but later (~ 1-16) we shall establish the expansion of
J, (kr cos 6 cos (D) J, (kr sin 4 sin 4>) by a direct transformation.

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