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

20

THEORY OF BESSEL FUNCTIONS

[CHAP. II

It is frequently convenient to modify (1) by bisecting the range of integration and writing 27r - for 0 in the latter part. This procedure gives
(2) Jn(z)=- cos (nO - z sin 0) dO.
7/ o
Since the integrand has period 2r, the first equation may be transformed
into
I 2r+a
(3) Jn (z) =2w fJ cos (rn - z sin 0) dO,
where a is any angle.
To prove (1), multiply the fundamental expansion of ~ 2'1 (1) by t-n-l and
integrate* round a contour which encircles the origin once counterclockwise.
We thus get
^. Y-nU -'I dt ^ rCtm-n- dt.
1 (0+) t -e -(t-l/t) p ()o i
27ri m =-X 2- Ji
The integrals on the right all vanish except the one for which n = n; and
so we obtain the formula
I j (0+) _ z(t-l/t) 1
(4) Jn ()= 27 t-i e At.
Take the contour to be a circle of unit radius and write t = e-it, so that 0
may be taken to decrease from 27r + a to a. It is thus found that
1 r2rf+a
(5) JJn (Z) = -. ei(n- z sin 0) d0,
a result given by Hansent in the case a = 0.
In this equation take a =- rT, bisect the range of integration and, in the
former part, replace 0 by - 0. This procedure gives
Jn(z)= 21 {ei(no-zsin 0) + e - i - z sin) dO,
and equation (2), from which (1) may be deduced, is now obvious.
Various modifications of Bessel's integral are obtainable by writing
n () = cos nO cos (z sin d - sin sin (z sin 0) dO.
7 0 7J 0
If 0 be replaced by wr - 0 in these two integrals, the former changes sign when
n is odd, the latter when n is even, the other being unaffected in each case;
and therefore
Jn (z) = - sin nO sin (z sin 0) dO
(6) 2 7r (n odd),
= — \ sin nO sin (z sin 0) dO
7rJo /
* Term-by-term integration is permitted because the expansion is uniformly convergent on the
contour. It is convenient to use the symbol J(a+) to denote integration round a contour encircling
the point a once counterclockwise.
t Ermittelung der absoluten Storungen (Gotha, 1843), p. 105.