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

134

THEORY OF BESSEL FUNCTIONS

[CHAP. V

Now apply this result to any two equations of the type of ~ 4'31 (17). If
5V, 9, denote any two cylinder functions of orders /, and v respectively, we have
(z) ( Z O' d) L[ l' dz - d (
[)"' (Z). Wm" to)i~ WI ~ Z i~' Z,
2 1{ (z) '(z) 1 (z) +. (Z) j ()
-[( 4 3 (z) + 102 (Z) _ 2 + 4} { (Z)
)'() + 4 "(z)) _ ())
x ). {+ (3zj V {1f (z)} W*2 (z) + dz),
where qb (z) and 4r (z) are arbitrary functions of z.
This formula is too general to be of practical use. As a special case, take
) (z) and * (z) to be multiples of z, say kz and Iz. It is then found that
(7) I {(k - 12)z - Z,$ (Z) 'i (z)dz
- = ( d( - W,(lZ) dR() - dQc (z)
dz dz
= {fa'+, (l,) < (z) - 1W (kz),+ (Iz)} - ( - v),, (kl) 9v (IZ).
The expression on the left simplifies still further in two special cases (i)u = v,
(ii) k = 1.
If we take, = v, it is found that
(8) () () { +, (kz) |, (i) - W, (dz) k + (s)1
This formula may be verified by differentiating the expression on the right.
It becomes nugatory when k = 1, for the denominator is then zero, while the
numerator is a constant.
If this constant is omitted, an application of l'Hospital's rule shews that,
when I - k,
(9) fz (l), (kz ), (k) dz = - 2 klz,+ (kz);' (kz)
- kzls (kz) ',+1 ( ( Z) ) +1 (z)}.
The result of using recurrence formulae to remove the derivates on the
right of (9) is
(10) fW (lkz) ',^ (kaz) dsz -= 1 2 (z) s2 (Ckz)( - (Wz) 1 (aZ)
- +1 (kza - (lkz)}.