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

7-25, 7-3]

ASYMPTOTIC EXPANSIONS

205

For large values of x, the difference between
y(v +n +, 2x)..
7(V+n 2x) and ( +v)n
is O( xv^n+ e-22) which is o (1) for each integral value of n.
In the case of the ordinary Bessel functions, we take the expression for
the function of the third kind
H/ 2\_ e(x- vrr-trr) 1 e (1 + iuV)H2) = (e = -) e A (I r ( ~ l ( + de
2x
so that
7 9 \ ~ir(X-VT-1n t) r2x t q q- If 2X
= f-) ---- ^ e-u + -} du + 0 (Xv e-2X)
7Tr X r(7) + 2xJ v /
2 ei 9 X \ - s 7r7r ) oo M I, 2
In2) ~ (7 2 wXt rem -),
and similarly
( i 2x) du
2(3) ( - (2 i(x- -) ( ) y)m, i (Pv +, ( 1, 2x)
(2) H/~) (x) = 2
afte ar c sto r (V a e Zpio m of (2ix)' )
nql(mertcatly 2s -a the first tr nget, ay a, sli more 2x)
77KX =_0) r (V + ~).m! (- 2ix)m
From these results it is easy to derive convergent series for the functions
of the first and second kinds.
Hadamard gave the formulae for functions of order zero only; but the extension to
functions of any order exceeding - - is obvious.
7'3. Formulae for the remainders in the asymptotic expansions.
In ~ 7'2 we gave an investigation which shewed that the remainders in
the asymptotic expansions of H (1) (z) and H,~ (z) are of the same order of'
magnitude as the first terms neglected. In the case of functions of the first
and second kinds, it is easy to obtain a more exact and rather remarkable
theorem to the effect that when v is real * and x is positive the remainders
after a certain stage in the asymptotic expansions of J~r (x) and Y~+ (x) are
numerically less than the first terms neglected, and, by a slightly more recondite investigation (~ 7'32), it can be proved that the remainders are of the
same sign as the first terms neglected.
Let us write
+2 Ke) uv i-+ + (e) - du =P(x, v),
2p(. + 1).V o +x;;i)\v1 / 2\
UJ-u- l ^ - -
2tiF(~1) fe t + dtu = tt(X, ),
* We may take v >0 without losing generality.