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

CHAPTER XVIII

SERIES OF FOURIER-BESSEL AND DINI
18'1. Fourier's formal expansion of an arbitrary function.
In his researches on the Theory of Conduction of Heat, Fourier* was led
to consider the expansion of an arbitrary function f(x) of a real variable of x
in the form
00
-(1) f (c)= Z amJo (jim),
m= 1
where j,, j2, j,... denote the positive zeros of Jo (z) arranged in ascending
order of magnitude.
The necessity of expanding an arbitrary function in this manner arises
also in Daniel Bernoulli's problem of a chain oscillating under gravity and in
Euler's problem of the vibrations of a circular membrane with an initial arbitrary symmetrical displacement (~~ 13, 1'5).
In order to determine the coefficients am in the expansion, Fourier multiplied both sides of (1) by x JO (jmx) and integrated between the limits 0 and 1.
It follows from ~ 5-11 that
xJO (j x) Jo (j*) dx= { j), n,
Jo ' l~O m) = n,
and hence Fourier inferred that
(2) a =J 2 tf(t) Jo (jmt) dt.
i J m (j 0
If we now change the significance of the symbols ji, so thatt j,,, j2s, j...
denote the positive zeros of the function J, (z), arranged in ascending order
of magnitude, then
(3) f () = Y amJ, (.jnx),
m=l
where
(4) an= J2 ru w tf (t) JL (jmt) dt.
V v+l (jm) J
This more general result was stated by Lommel+; but, of course, neither
in the general case nor in the special case = 0 does the procedure which has
been indicated establish the validity of the expansion; it merely indicates how
the coefficients are to be determined on the hypothesis that the expansion
exists and is uniformly convergent.
* La TMhorie Analytique de la Chaleur (Paris, 1822), ~~ 316-319.
t The omission of the suffix v, associated with jl, j2, j..., should cause no confusion, and it
considerably improves the appearance of the formulae.
+ Studien iiber die Bessel'schen Functionen (Leipzig, 1868), pp. 69-73.