214 FURTHER APPLICATIONS OF BESSEI/S FUNCTIONS OF [380
It may be expressed in the form
which, however, requires a special evaluation when n is an integer. Using Schlafli's formula
Jn (*) = - I" cos (z sin 6 - nff) dd - -— f e~^ ^ e d6, . . .(12)
7T J o T J o
n being positive or negative, and z positive, we find
6~* sinl1 e d0 + — — f e-»6-z sillh e
T Jo
- - f* sin (2 sin d-nB)dO-- f ^ cos (2 sin 0 - n8) d6, ...... (13)
TrJo TrJo
the imaginary part being - iJn (z} simply. This holds good for any integral value of n. The present problem requires the examination of the form assumed by Dn when n is very great and the ratio zjn decidedly greater, or decidedly less, than unity.
In the former case we set n — z sin a, and the important part of Dn arises from the two integrals last written. It appears* that
r+, ........................ (14)
rrz cos a.)
where p ~ |TT + z {cos a - QTT - a) sin a}, .................. (15)
or when z is extremely large (a = 0)
Or+«) ......................... (16)
(17)
^ '
V7T37
At a great distance the value of <£ in (9) thus reduces to
--\7rkr
from which finally the imaginary part may be omitted.
When on the other hand zjn is decidedly less than unity, the most important part of (13) arises from the first and last integrals. We set n = zcosh/3, and then, n being very great,
where t = n (tanh j3 - /3) ............................ (19)
* Nicholson, B. A. Report, Dublin, 1908, p. 595 ; Phil. Mag. Vol. xix. p. 240 (1910) ; Mac-donald, Phil. Trans. Vol. ccx. p. 135 (1909). •ndrical surface at r = a, and it turns upon the character of the function of r which represents a disturbance propagated outwards. If Dn (kr) denote this function, we have