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

164

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

Thus, when R (z) > 0 and v + - is not a positive integer,
P 1 7\ eviri (1 Z\v (-I+,1+)
(2) J- (z) (-) evr ( ), + eiz (t2 " 1)v- dt.
This equation was also obtained by Hankel.
Next consider
(-i+,) eizt (t2 1)V- dt,
co exp( -io)
where o is an acute angle, positive or negative. This integral defines a function
of z which is analytic when
- 7r + o < arg z< 7r + )o;
and, if z is subject to the further condition that arg z < -7, the contour can
be deformed into the second of the two contours just considered. Hence the
analytic continuation of J- (z) can be defined by the new integral over an
extended range of values of arg z; so that we have
J_ ()- (- zJ) eVri (Iv (-( -+,1 +)i
(3) ~J-_.)- (22 v) 2 ) izt (t2- _)v27i 1 (I) J i exp(-io)
where arg z has any value between - -r + wo and 7r + co.
By giving () a suitable value*, we can obtain a representation of J_- (z)
for any assigned value of arg z between - r and t-.
When R (z)> 0 and R (v + ~) > 0 we may take the contour to be that shewn in Fig. 3,
-o -
Fig. 3.
in which it is supposed that the radii of the circles are ultimately made indefinitely small.
By taking each straight line in the contour separately, we get
= 2) _ -e (Z)- f eizte-v3i-(1 -t2)v- -dt
+iz e- 37i(v-2) ( -t2)v-c dt
J To
+ |eizt e-iv- 2) (l-t2)v - dt
-1
+ eizt eri(rv-) (I t2)v- 1-dt
-J i
~ eit enri(v) (1J- t2)v-^2dt
0~~~

* If w I be increased in a series of stages to an appropriate value (greater than ~rr), a representation of J-v (z) valid for any preassigned value of arg z may be obtained.