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

432

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

It is also supposed that | arg k I < wr, and the loops in the contour surround
the points
eai k /2, ei k,/2.
By analysis resembling that of ~ 13'52, the reader will find that
(2) [cos ( p- Iv- 2j) v. J,(ax) + sin ( p- - - 2- ) 7. Y, (ax)] ( + 4k+ d
7r (k 2)P-o-4 [/ (ak/'/2)v+2nm r (p+ ~ + jV m) p+v-4,+2m
2sin v7r.r(/+1) Lm=0= n!r(v+m+ l)Fr(p+~v-, +-?m)s 4
(aCk/22)-V+2m rF.(p - 4 v + cos p - v - 4 + 2m n]
4- 4;2 ) C
mo=0! r (-v + m + 1)F r ( p- v - + n) 4 j
If the series on the right are compared with those given in ~ 5'41, it is seen
that the former is expressible as a product of Bessel functions if p - 2 = = /, + 2
or if p - 4 = v= =/ + ~, while the latter is so expressible if p - 2 = - Uv=/ +
or if p-4 - = =/ +-.
The corresponding integral which contains a single Bessel function will be
considered in ~ 13'6.
13*55. Sonine's integrals.
A number of definite integrals, of which special forms were given by
Sonine, Math. Ann. xvI. (1880), pp. 63-66, can be evaluated by the method
of contour integration.
The most general contour integral to be taken is
2r1 i ei(z+k)
X (+ k)mr+l (- Z)"-' Hvl) (- z) d
round a contour consisting of the parts of the circles
z, =8, Izl=R,
terminated by the lines arg (- z) = + 7r, and the lines which join the extremities
of these circular arcs*.
It is supposed that m is an integer and k is not a negative real number.
The integral round i z = 8 tends to zero as 8-0, provided that R (p) > R (v) l,
and the integral round I z = R tends to zero as R - oo, provided that
R(p)< + 3.
By Cauchy's theorem we have
ei (z+k) ()meik dmr
\ (z7 e k)(- Z)en - Aly(1) ( Z) dz = -7-e- ik p ()
* Cf. Modern Analysis, ~ 6-2; or ~ 7 4 supra.