416 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
§ 5 . The best known example of the fourth type is Bessel’s function 
~ \2/a + 
- 
J ( z ) — X -—A-. 
^ ’ f =Q r (/i + i) r (/i + 7i + 1 ) 
It is evident that z - " J„ ( 2 ) is a uniform integral function. 
The investigations of Poisson, # Stokes, t Lipschitz,| and Jordan, § have finally 
led to a rigorous demonstration by the latter that asymptotically, when n is real, 
z n J„ (z ) = \J z “ 1 cos jz — (n + J) [, when Tv ( 2 ) is positive, 
and z~ n J„ (:) = \/~~ e '" + 1 ' ? cos jz + (71 -f- |-) q 1, when T\ (z) is negative. 
The complexity of this result is reduced by the transformation — z 2 = t or z — 1 \/t, 
00 t ]X 
hich gives (z) = E ^ _ . - .. _ . -—, an integral function of t. 
h ny ' „ =0 2^+»r( /i + i)r(g + 7i + ly 6 
And now we have for the asymptotic value of z~"J n ( z ) the unique expression 
, 2n + 1 1 (...) 
( 2 n)~- t~ 4 e l ~ + t + 
which is valid for all values of arg t between — jt and tt. 
This shows at once that z“"J„ (z), qua function of t, has no imaginary roots which 
are not at a finite distance from the negative part of the real axis. In point of 
fact, these roots are known to be real and negative when n > — 1 ;|. Hence the 
asymptotic expansion for 
t~ 2 
e 2 J„ ( e2 
x> 
m=o 
2 2 > j - + n r(/i + i) r(g + n + i) 
is valid for all points in the neighbourhood of t = 00 except those which are at 
a finite distance from the zeros of the function. 
§ 6. The question now forces itself upon us :—“ Do all integral functions of a 
single variable z admit asymptotic approximations in the. domain of z— 00 , which are 
valid for all 'points but those which are in the immediate vicinity of the zeros of the 
functions ? 
* Poisson, •Journal cle l’Ecole Polyt.,’ vol 19, 1823, pp. 349 d seq. 
t Stokes, ‘Camb. Phil. Trans.,’ vol. 9, 1856, pp. 166 et seq. 
X Lipschitz, ‘ Crelle,’ vol. 56, pp. 189 et seq. 
§ Jordan, ‘Cours d’Analyse,’ 1896, vol. 3, pp. 254-274. 
| When u is negative and between m and m 4 - 1 in absolute value, there may be a finite number (2m) 
of imaginary roots of z~ n J, t (z), but these are not associated with the essential singularity. Cf. Mac¬ 
donald, ‘ Proc- Lond. Math. Soc.,’ vol. 29, pp. 575-584. 
