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

360

THEORY OF BESSEL FUNCTIONS

[CHAP. XI

Graf's proof is based on the theory of contour integration, but, two years after it was
published, an independent proof was given by G. T. Walker, Messenger, xxv. (1896), pp. 76 -80; this proof is applicable to functions of integral order only, and it may be obtained
from Graf's proof by replacing the contour integrals by definite integrals.
To prove the general formula, observe that the series on the right in (1)
is convergent in the circumstances postulated, and so, if arg Z= a, we have
J^+m (Z) Jm () eri
M = -- o
2t6z= 2w (o+) exp Z t — ) t —m- J_, (z) emiOdt
2TZi m=-c Jo -oo exp(-ia) t/)
1 (+)\ t ei^ dt
2ri J00 exp(-ia) exp - t-t 2 e t;
there is no special difficulty in interchanging the order of summation and
integration.
Now write
(Z - ze-i ) t = su, (Z- zei4)/t = /u,
where, as usual, r = \/(Z2 + z2 - 2Zz cos 0 ),
and it is supposed now that that value of the square root is taken which makes
-+ Z when z - 0.
For all admissible values of z, the phase of w/Z is now an acute angle,
positive or negative. This determination of v renders it possible to take the
u- contour to start from and end at - oo exp (- i/), where 83 = arg w.
We then have
1 (Z-ze-i^ O+) ( ) du
Jn+m (Z) Jm (z) ei4 - i =- exp {z (U - -
M1= -o 27 ri s -- ooexp (-i)u
Z - ze-i V^
\ Z-ze-k ] J (a),
by ~ 6-2 (2); and this is Graf's result.
If we define the angle / by the equations
Z-zcosP os, co sin, z sin = sin,
where 4 - 0 as z 0 (so that, for real values of the variables, we obtain the
relation indicated by Fig. 28), then Graf's formula may be written
oo
(2) eviq J, (y) = E Jv+m (Z) Jm (z) en4i,
nz=-X
and, on changing the signs of g and it, we have
c(3) e-vi J ( ) = t J$,,+ (Z) J$ (z) e-m
Mi= -ao

* Of. Bromwich. Theory of Infinite Series, ~ 176.