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Full text of "Scientific Papers - Vi"

4                       NOTE ON BESSEL'S FUNCTIONS AS APPLIED
since by (3)
»   n) +   n  n = - n In (5) all the denominators are positive.    We deduce
„ 2 _ ™2                      ,?• 2 _ OT2         -. 2 _ ™2
gl           ^   __  1     ,   *1 _ 11    I    ^           ^      ,               ^ I    •
*
and therefore
z? >n2 + '2n>n(n + 1).
Our theorems are therefore proved.
If a closer approximation to z-f is desired, it may be obtaine stituting on the right of (6) 2n for z? — n2 in the numerators and i n2 in the denominators. Thus
r\          -•'   J-   T     •*•""
2n
•^   + 2« ,,«!    , «2    « *s    i ...    ,„/»,~L~O\ r •
ft ^l +• 4; J
Now, as is easily proved from the ascending series for Jn',
71+2
ty  ~"2   _J_   /y   "™*2      1_    «•   "~~2   _l               —                                       •
so that finally
w3
z? >n-+ 2-» +,-—
When n is very great, it will follow from (7) that z? > v/,2 + the approximation is not close, for the ultimate form is*
^ = 7is+[1-6130] n4*
As has been mentioned, the sequence formula In
Jn (g) = Jn_1 (z] + J
n+l
prohibits the simultaneous evanescence of J"n_1 and Jn, or of Jn-i The question arises—can Bessel's functions whose orders (supposei differ by more than 2 vanish simultaneously ?    If we change n in (8) and then eliminate Jn, we get
_
- "«— 1 ~1          '"
tl+2>
from which it appears that if Jn^ and Jn+2 vanish simultaneously, t Jn+i = 0, which is impossible, or ^-' = 4<n (n + 1). Any common rt and Jn+i must therefore be such that its square is an integer.
* Phil. Mag. Vol. xx. p. 1003, 1910, equation (8).    [1913.   A correction is hen See Nicholson, Phil. Mag. Vol. xxv. p. 200, 1913.]emains to prove that z* necessarily exceeds n(n 4-1). That zli exceeds n2 is well known*, but this does not suffice. We can obtain what we require from a formula given, in Theory of Sound, 2nd ed. § 339. If the finite roots taken in order be zi} zz,... za..., we may write