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

236

THEORY OF BESSEL FUNCTIONS

[CHAP. VIII

The contour has now been selected* so that the integrand does not
oscillate rapidly on it; and so we may expect that an approximate value of
the integral will be determined from a consideration of the integrand in the
neighbourhood of the pass: from the physical point of view, we have evaded
the interference effects (cf. ~ 8'2) which occur with any other type of contour.
The mode of derivation of asymptotic expansions from the integral will be
seen clearly from the special functions which will be studied in %~ 8'4-8'43,
8'6, 8'61; but it is convenient to enunciate at this stage a lemmat which will
be useful subsequently in proving that the expansions which will be obtained
are asymptotic in the sense of Poincare.
LEMMA. Let F(r) be analytic when I T I a + 8, where a > 0, 8 > 0; and let
F(r)= E an7(m/r)/-1,
m = 1
when iT a, r being positive; also, let IF (7) < Keb, where K and b are
positive numbers independent of T, when 7 is positive and r > a. Then the
asymptotic expansion
e-vrF(r) dr - ' ar (m/r) v-nl/r
JO m=1
is valid in the sense of Poincare when v is sufficiently large ancj
arg v < r- A,
where A is an arbitrary positive number.
It is evident that, if M be any fixed integer, a constant K1 can be found
such that
F () - ~ ar (m/r)-1 < K1T(M/r)-1eb
= 1
whenever 7 > 0 whether v < a or T > a; and therefore
e-'TF ((T) d7 = j e-vramr(mT/)-1T d + R1,
hr
where R, < e-"~. K, (Mlr)-l ebr d
= K r (M/r)/IR (v) - b}M/
= (0p-J/r),
provided that R (v) > b, which is the case when |v I > b cosec A. The analysis
remains valid even when b is a function of v such that R (v)- b is not small
compared with v. We have therefore proved that
je-vF (r)dr = E amr (m/r) v-mr + 0 (V-3r),
and so the lemma is established.
* For an account of researches in which the contour is the real axis see pp. 1343-1350 of
Burkhardt's article in the Encyclopddie der Math. Wiss. II. 1 (1916).
t Cf. Proc. London Math. Soc. (2) xvI. (1918), p. 133.