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

26

THEORY OF BESSEL FUNCTIONS

[CHAP. II

and hence, on integration, when n > 1,
(2n - 1) cos (z cos 0) sin221-2 Od - 2n cos (z cos 0) sin21 t dO
Jo o
+ 2m+ 4J cos (z cos 0) sin22l+2 OdO 0.
2n+1 o
If now we write
1r (nI + ) > cos (z cos 0) sin2' OdO - (n),
the last formula shews that
z - (n - 1) - 2nf (n) + (n + 1) = 0,
so that b (n) and J,,(z) satisfy the same recurrence formula.
But, by using Bessel's integral, it is evident that: (0)= Jo (z),
(1) = cos ( cos )si cos) sin d 0d08
1 sin (z cos os cos = - Jo' (z) = J (),
7/'.0
and so, by induction from the recurrence formula, we have
s (dn) = Jn, (Z),
when in = 0, 1, 2, 3,....
2'32. Jacobi's investigation of Poisson's integral.
The problem of the direct transformation of Poisson's integral into Bessel's
integral was successfully attacked by Jacobi *; this method necessitates the use
of Jacobi's transformation formula
di- sin2- 0 () 1. 3..5... (2n- 1)
d = ()l -sin n,
where / = cos 0. We shall assume this formula for the moment, and, since no
simple direct proof of it seems to have been previously published, we shall
give an account of various proofs in ~H 2'321-2'323.
If we observe that the first n - 1 derivates of (1 -/2)1-2, with respect to
/j, vanish when i = ~ 1, it is evident that, by n partial integrations, we have
zlI cos (z os s 0) sin2 Odd = z1 cos (z/). (1 - 2)L- db,
)-1
-- (-()n1 cos (z - n)d (1- ) d-.
* Journal fir Math. xv. (1836), pp. 12-13. [Ges. Math. Werke, vi. (1891), pp. 101-102.] See
also Jo7urnal de Math. I. (1836), pp. 195 —196.