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

19-22]

-- SCHLOMILCH SERIES

625

19'22.. The construction of the function F (z).
We now take the contour integral
1 (+)F(z) -—.-^ (-) dz,
27rti Sin Trz
and, in order to calculate it in a simple manner, we shall suppose that F(z) is
expansible in a series of Filon's type
(1) F(z)= pn k, (z)
= 1 Zn+1
1n=l zil
where 4n (z) denotes the sum of those terms in the expansion of 7r-l sin wrz
whose degree does not exceed n, and the coefficients pm will be defined later.
The reader will observe that
+1 (z) = 2 (Z)=
3 (Z) = 4 (z)=z- 23,...............................
With this definition of F (z), it is evident that, for small values of z z,
F /(Z) v7,(xz) P {- 7r*,, (z), (
sin 7Z = P nl1 sin ( -z (
4 1 r-n+ cos T +.; 0 (Z))
n=1 1sn+l (n+ 1)! sin 7rz ) ()
It follows immediately that
/b(oco p rn COS p (_____:l (n= +) i n=1.2.r( iir.+ )
and consequently we proceed to identify
_oo p (:.. r (r(in+I) r(~,,?{ +!),
with f(x) -f(O). For this purpose we have to assume temporarily thatf (x)
has differential coefficients of all orders at the origin, and then we define the
coefficients p, by the equation
f/n (0) = Pn (Q i)n
(3)Mn! — (,.+) l( +,+l)' (n=1,2,3,...).
We next transform this equation defining pn in such a way that the sum
of the series, by which F(z) is defined, is expressible in a compact symbolic
formn; the transformation of the series for F(z) can be effected by expressing
* This type of series is fundamental in Filon's theory, and is not peculiar to Schlomilch
expansions; thus, in his work on Fourier-Bessel series, sin 7rz is replaced by z-VJ, (z) and (z)
denotes the sum of the terms whose degree does not exceed n in the expansion of that function.

W. B. F.

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