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

282

THEORY OF BESSEL FUNCTIONS

[CHAP. IX

" e - u du
The Borel-sum associated with this series is (1 + t2) (1 -t2) - 2tz' and this integral
is convergent so long as (1 - t2) z/t is not negative.
There is no great difficulty in verifying that the series E (-) ent"On (z) is an asym9l=0
ptotic expansion of the integral for small positive values of t when arg z < 7r, and so the
integral may be regarded as the generating function of 0n (z). Kapteyn has built up much
of the theory of Neumann's function from this result.
9 17. The inequality of Kapteyn's type for On (n2).
It is possible to deduce from Neumann's integral an inequality satisfied
by On (nz) which closely resembles the inequality satisfied by J, (nz) obtained
in ~ 8'7.
We have
On (nz) = nlo + (W + Z()}W + {w - /(w2 + Z2)}n] e-nw dw,
the path of integration being a contour in the w-plane, and so
On (nz) 1 n+l[{w + /2+ z2) e-"] I d
where that value of the radical is taken which gives the integrand with the
greater modulus.
Now the stationary point of
+ ( + V( 2) e-w
is V(1 - Z2), and so
(1) ~On (nz) 1 (1 _ 2) ( {w+ V(w2 + 2z)} e-w. I dw 1,
where the path of integration is one for which the integrand is greatest at the
stationary point.
If a surface of the type indicated in ~ 8'3 is constructed over the w-plane,
the stationary point is the only pass on the surface; and both w = 0 and
w = + co are at a lower level than the pass if
z exp /(1 - z <) 1
(2) <
1 + 4(1- z2)
Hence, since a contour joining the origin to infinity can be drawn when (2) is
satisfied, and since the integral involved in (1) is convergent with this contour,
it follows that, throughout the domain in which (2) is satisfied, the inequality
A 1 + V(1 - z2) n-,
(3) (nz) < IZX 12 'z exp /(1 - ')
is satisfied for some constant value of A; and this is an inequality of the same
character as the inequality of ~ 8'7.