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

INTEGRAL REPRESENTATIONS

169

and hence that
m o( J( 2v+1Zv T7 rcos- 0. sin (z- v + 20),-2ZCot dO,
(7) Y( ) r ( -) r ()o sin2+10
2 +0 -Ccos 0 -Pcos(- + ~0) eCot0
(8) L %( = - --------- r (L )f-2V- s e. do.
These formulae, which are of course valid only when R (v + ) > 0, were
applied by Schafheitlin to obtain properties of the zeros of Bessel functions
(~ 15'32-15'35). They were obtained by him from the consideration that the
expressions on the right are solutions of Bessel's equation which behave in the
appropriate manner near the origin.
The integral f e- t-v (1 + u)f- diu, which is reducible to integrals of the types
occurring in (3) and (4) when A = v, has been studied in some detail by Nielsen, Math. Ann.
LIX. (1904), pp. 89-102.
The integrals of this section are also discussed from the aspect of the theory of asymptotic solutions of differential equations by Brajtzew, Warschau Polyt. Inst. Nach. 1902,
nos. 1, 2 [Jahrbuch iiber die Fortschritte der Math. 1903, pp. 575-577].
6'13. The generalised Mehler-Sonine integrals.
Some elegant definite integrals may be obtained to represent Bessel functions
of a positive variable of a suitably restricted order. To construct them, observe
that, when z is positive (= x) and the real part of v is less than 1, it is permissible to take o = }7r in ~ 611 (6) and to take o =- 7Irr in ~ 6'11 (7), so
that the contours are those shewn in Fig. 6. When, in addition, the real part
of v is greater than - 2, it is permissible to deform the contours (after the
manner of ~ 6'12) so that the first contour consists of the real axis from + 1
to + oo taken twice while the second contour consists of the real axis from
- 1 to - oo taken twice.
Fig. 6.
We thus obtain the formulae
1(fj (a) = (1-V ) (i _ e-2(V-)) fxt (t2 - 1) dt
~l (it - 1 - d,
lH (2) ()- r ) ( (1 - e2 (v-2i ) e-izt (t2 - 1)V- dt,
the second being derived from () by replacing t by t.
the second being derived from ~ 6'11 (7) by replacing t by -t.