be applied in this paper. 
The conditions which will be imposed are the following. 
First, the functions, f(x) and its first derivative, f'(x), are 
continuous as stated in (8.6). Secondly, the absolute value 
of f(x) is less than some positive constant M as stated by equa- 
tion (8.7). And thirdly, the absolute value of the derivative 
of f(x) is less than M. 
If these conditions are satisfied, then it can be proved 
that equation (8.9) holds. The proof follows. The integral 
from -A to A can be broken up into three parts as in equation 
(8.10) where «€ is some small but fixed number. The integral from 
-A to -e and the integral from e to A can be shown to vanish. 
The integral from - « to € contributes the whole value to the 
entire integral. 
Consider, first, the integral from ¢«€ to A. It can be in- 
tegrated by parts, and if absolute values are taken, the first 
inequality in equation (8.11) is the result. Estimates based 
upon M, the value of « , and the length of the path of integra- 
tion, then yield the second inequality. The second inequality 
as N approaches infinity tends toward zero. Therefore the inte- 
grals from € to A and from -A to - e tend to zero as N approaches 
infinity. 
Consider next, the integral from -¢« toe . In equation 
(8.12), the transformation of variables given by equation (8.13) 
yields the first expression. As n approaches infinity f(x'/N) 
approaches f(o) over a large range of x', and f(o) can be factored 
out of the integral as a constant. The integral from - N to 
N of (sin x')/x' approaches the integral from minus infinity 
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