1914-15.] Expansions of the Interpolation-Theory. 
185 
imaginary infinite values of the argument are supposed to have been 
removed already by the method of § 2. Let g{x, k) denote the function 
—J /(/x) J 6 cos X{x - fx)dX . 
Here k denotes a positive constant, introduced for the purpose of securing 
convergence in the following developments. 
Break up the range of integration in g{x, k), thus — 
g(x, k) = — 
^00 
77 StT StT 
^ /»- 
1 W 1 w 1 w 
+ + + — 
Jo 
— IV w 
e cos X{x - ix)dX . 
The first partial integral consists of terms whose period in x is greater 
than 2w, the second partial integral consists of terms whose periods are 
between 2w and f n;, and so on. Replace every periodic term whose period 
is less than 2w by the corresponding cotabular term whose period is greater 
than 2w, as explained in the preceding article. We thus obtain an 
expression which we shall denote by G(x, k), where 
7 T 7-00 
W ^-Kk 
e cos {X(x — fx)}dX 
I. 
+ e ^cos I X{x - [X) + —(a - /x) | o'A. 
~ w 
+ ^(^^«’)cos| - /x) + — (a - /x) | (iX 
+ 
Summing the series of exponentials and cosines, we have 
,5 . , ^irk 
w smn cos 
^ - Afc w 
I X{x - /x) I - sin I X{x - ft) j> sill j — (a - /x) j- 
- 7T 
W 
cosh 
‘hrk 
dX 
cos — (a - ix) 
w 
Performing the integration with respect to X, this gives 
X) = 
27 TZ 
d(xf{p) 
sin - (a; - /X - ik) cos - ft - ik) sir -(x- ul + ik) cos - (a - ft + ik) 
10^ w IV w 
(x- jjL- ik) sin ^ (a - ft - ik) 
{x- jx + ik) sin — (a - ft + ik) 
Now if we evaluate the integral 
^iri 
-(x-fj.-ik) COS —(a - ft - ik) 
dfxf{fX); 
w 
ft - ik 
sin —(a-fx- ik) 
%(! 
