183 
1914-15.] Expansions of the Interpolation-Theory. 
places a, a + w, a — iv, etc. : and this function has no singularity at c, since 
the infinite part of the term 
. Trix — a) 
r sin — ^ ^ 
{x - c) sin 
-(c - a) 
exactly neutralises the infinite part of f{x). Moreover, this term does not 
introduce any fresh singularity in the finite part of the ic-plane, and does 
not cause the new function to become infinite even at x = co so long as x 
is real. 
This establishes the result for the case when the singularity is a simple 
pole. When it is a pole of higher order, or an essential singularity, we 
can make use of the known result that the part of the expansion of f{x) 
which becomes infinite near this singularity may be expressed in the form 
f /(zyh 
27ri I Z — X 
Jy 
where y denotes a small circle enclosing the singularity c. Now this can 
be neutralised by a term 
f{z)dz 
1 . 7t(x - a) 
— . sm ^ ' 
2tvi 
w I {z — x) sin 
'{z - a) ’ 
and as this term contains sin as a factor, it vanishes when the 
. . Hence in this 
w 
argument has any of the values a, a-\-w, a — w, u + 2w, 
case also we can write down a function, namely, 
f{z)dz 
+ 
sm 
■(x — a) 
\-ni 
{z - x) sin 
7t{z - a) ’ 
which is cotabular with f{x) but has no singularity at the point x = c. 
By repeated application of this process we can remove all the singu- 
larities of f(x) in the finite part of the plane, and obtain a function 
which is cotabular with f(x), and which does not become infinite except 
for values of x whose imaginary part is infinite. 
§ 3. Removal of rapid oscillations from a function, by substituting 
a cotabular function for it. 
Having replaced the original function fix) by a cotabular function of 
the kind just described, we shall now suppose the latter function to be 
analysed into periodic constituents by Fourier’s integral-theorem (or, in 
