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

19-2]

]SCHLOMILCH SERIES

621

is uniformly summable (C 1) throughout (0, r.); and hence, by term-by-term
integration, the series
ao + - a JO (mx)
12Z =1
is uniformly summable (C 1) throughout (0, 7r), with sum f(x); an application
of Hardy's convergence theorem* then shews that the additional condition
am = o(l//m)
is sufficient to ensure the convergence of the Schlomilch series to the sum
f(x) when x lies in the half-open interval in which 0 < x <, r.
For further theorems concerning the summability of Schl6milch series,
the reader should consult a memoir by Chapmant.
[NOTE. The integral equation connecting f(x) and g (x) is one which was solved in 1823
by Abel, Journal fiir Math. I. (1826), p. 153. It has subsequently been investigated by
Beltrami, Ist. Lombardo Rendiconti, (2) xIII. (1880), pp. 327, 402; Volterra, Ann. di Mat.
(2) xxv. (1897), p. 104; C. E. Smith, Trans. American Math. Sod. vII. (1907), pp. 92-106.
The equation
- m s f:f' (X sin 0 sin 4)" sin 0 d dO =f (x) - f(O)
is most simply established by the method of changing axes of polar coordinates, explained
in ~ 3'33; this method was used by Gwyther, Messenger, xxxIII. (1904), pp. 97-107, but
in view of the arbitrary character of f(x) the analytical proof given in the text seems preferable. i In connexion with the changes in the order of the integrations, cf. Modern Analysis,
~ 4-51.]
19'2. The definition of Schlomilch series.
We have now investigated Schlomilch's problem of expanding an arbitrary
function into a series of Bessel functions of order zero, the argument of the
function in the (m + 1)th term being proportional to m; and the expansion is
valid for the range of values (0, 7r) of the variable.
Such series may be generalised by replacing the functions of order zero by
functions of arbitrary order v; and a further generalisation may be effected by
taking the general term to contain not only the function J, (nmx) but also a
function which bears to the Bessel function the same kind of relation as the
sine does to the cosine. The latter generalisation is, of course, suggested by
the theory of Fourier series, and we are thus led to expect the existence of
expansions valid for the range of values (- w, 7r) of the variable.
The functions which naturally come under consideration for insertion are
Cf. Modern Analysis, ~ 8-5.
t Quarterly Journal, XLIII. (1911), p. 34.
+ Some interesting applications of Fourier's integral theorem to the integral equation have
been made by Steam, Quarterly Journal, xviI. (1880), pp. 90-104.