those used in equations (10.1) to (10.22). 
How does E(# )*differ from E(w)? E(-)* is continuous, 
and it has the same value as E(#) at each point. But it has 
a derivative at no point, because the slopes at two neighboring 
points can be completely different. 
The above considerations are admittedly very crude explana- 
tions of what are, in reality, very complex properties of some of 
the more abstract functions treated in the theory of functions of 
a real variable. A study of the derivation given above from equa- 
tions (10.1) to (10.22) shows that actually functions like E(# )* and 
[ach )]o* are approached instead of functions like E(p) and 
PAE ue. 
However, and this is the important point, it is impossible 
to tell the difference between the starred and unstarred functions 
by any numerical or physical (electronic and/or mechanical) method 
of analysis of the original wave record. In any numerical method, 
a finite net must be taken, and the abstract differences between 
the functions cannot be detected. In an electronic or mechanical 
method of analysis, at some time in the analysis the record is 
sent through a tuned circuit of finite band width and again the 
abstract differences between the functions cannot be detected. 
For a further consideration along these lines, the papers of Tukey 
[1949] and Tukey and Hamming [1949] can be consulted. 
In conclusion, then, it is possible to think of the power 
spectrum and the cumulative power distribution as nice, smooth, 
elementary functions, to work with them as such, and differentiate, 
integrate, and transform them as such. With these considerations 
in mind, equations (10.22), (10.23) and (10.24) can be used with 
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