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

2'1, 2'11]

-THE BESSEL COEFFICIENTS

15

If in (1) we write - 1/t for t, we get
ez(-1lt+t)= | (- )-nJ.( )
00:= ( - )" J-_ ),
n= -C0
on replacing n by - n. Since the Laurent expansion of a function is unique*,
a comparison of this formula with (1) shews that
(2) J_9 (z) = (-)n Jn (z)
where n is any integer - a formula derived by Bessel from his definition of
Jn (z) as an integral.
From (2) it is evident that (1) may be written in the form
(3) e(t-/t) = J, (z) + tt + ( - )n t-) Jt (Z).
n= 1
A summary of elementary results concerning J. (z) has been given by Hall, The Analyst,
I. (1874), pp. 81-84, and an account of elementary applications of these functions to
problems of Mathematical Physics has been compiled by Harris, American Journal of
Math. xxxiv. (1912), pp. 391-420.
The function of order unity has been encountered by TurriBre, Nzouv. Ann. de Jfath. (4)
ix. (1909), pp. 433-441, in connexion with the steepest curves on the surface z=y (5x2 -y4).
2'11. The ascending series for J (z).
An explicit expression for J, (z) in the form of an ascending series of powers
of z is obtainable by considering the series for exp ( zt) and exp (-2 z/t), thus
00 (I (Y')'r Go I / 1 -M
exp t (t-1/0t)}= 2( (-= ) t-m
r r! =0 m'
When n is a positive integer or zero, the only term of the first series on the
right which, when associated with the general term of the second series gives
rise to a term involving t' is the term for which r = n + m; and, since n > 0,
there is always one term for which r has this value. On associating these
terms for all the values of m, we see that the coefficient of tn in the product is
0 (l Z)?+m (-,=o (n+mn) m In!
We therefore have the result
~(1) 37,(~) c= ( _ \m 1 ) )n+2m
n,=0o i1!(n + m)!
* For, if not, zero could be expanded into a Laurent series in t, in which some of the
coefficients (say, in particular, that of tm) were not zero. If we then multiplied the expansion by
t-m-1 and integrated it round a circle with centre at the origin, we should obtain a contradiction.
This result was noticed by Cauchy, Conptes Rendus, xIII. (1841), p. 911.