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

578

THEORY OF BESSEL FUNCTIONS

[CHAP. XVIII

It was pointed out by Dini that the expansion (5) must be modified* by
the insertion of an initial term when H + v = 0; and, although Dini's analysis
contains a numerical error, this discovery seems to make it advisable to
associate Dini's name rather than Fourier's with the expansion.
The researches which have now been described depend ultimately on a set
of lemmas which are proved by Cauchy's theory of residues. The use of complex variables has, however, been abandoned, so far as possible, by Kilesert
and Hobson+, who have constructed the expansion by using the theory of
integral equations as a basis.
On aesthetic grounds there is a great deal to be said for this procedure,
because it seems somewhat unnatural to use complex variables in proving
theorems which are essentially theorems concerning functions of real variables.
On the other hand, researches based on the theory of integral equations are
liable to give rise to uneasy feelings of suspicion in the mind of the ultraorthodox mathematician.
The theory has recently been made distinctly more complete by the
important memoir of W. H. Young~, who has thrown new light on many
parts of the subject by using modern knowledge of the theory of functions of
real variables in conjunction with the calculus of residues. An earlier paper
by Filonlj which makes some parts of the analysis appreciably less synthetic
must also be mentioned here.
The question of the permissibility of term-by-term differentiation of the
expansion which represents a function as a series of Bessel functions has been
discussed by Ford~, who has obtained important results with the help of
quite simple analysis (cf. ~ 18'4).
More recondite investigations are due to C. N. Moore"*, who, after studying
the summability of the expansion by Ceshro's means, has investigated the
uniformity of the convergence of the expansion in the neighbourhood of the
origin, and also the uniformity of the summability of the expansion (when
not necessarily convergent) in this neighbourhood.
The reason why the uniformity of the convergence (or summability) of
the expansion in the neighbourhood of the origin needs rather special consideration is that it is necessary to use asymptotic formulae for J, (Xx) which
are valid when )Xm is large; and, as x approaches zero, the smallest value of
m, for which the asymptotic formulae are significant, is continually increasing.
* Details of necessary modifications when H+v < 0 will be given in ~ 183. The modification
was also noticed by Kirchhoff, Berliner Sitzungsberichte, 1883, pp. 519-524.
t Archiv derMath. und Phys. (3)vnI. (1903), pp. 123-133; Math. Ann. LXIII. (1907), pp. 477-524-. Proc. London Math. Soc. (2) vii. (1909), pp. 359-388.
~ Ibid. (2) xvII. (1920), pp. 163-200.
1 Ibid. (2) iv. (1906), pp. 396-430. Cf. ~~ 19 21-19-24.
T Trans. American Math. Soc. iv. (1903), pp. 178-184.
- Ibid. x. (1909), pp. 391-435; xnI. (1911), pp. 181 —206; xxi. (1920), pp. 107-156.