PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 641 



It is the purpose of this paper to take the first steps towards the 

 preparation of two tables, one giving the evaluations of Fourier 

 integrals and the other giving the sinusoidal solutions for physical 

 systems. Together they would reduce the practical application of 

 the Fourier integral to the selection of three results from these two 

 tables. Thus by means of the first table the arbitrary cause could be 

 resolved into a sum of sinusoidal causes ; by means of the second table 

 the solutions for these sinusoidal causes could be supplied; and, 

 finally, by means of the first table again, the effect of superposing 

 these sinusoidal solutions could be shown, and thus the answer to the 

 original problem would be given. 



The preparation of the tables calls primarily for a compilation of the 

 results already obtained by pure analysis, after which new evaluations 

 and new solutions should be added, in so far as is possible. No attempt 

 has yet been made to completely cover the existing literature on the 

 subject, which extends back over one hundred years and is extensive 

 and widely scattered. But sufficient has been done to show that the 

 forms of the tables which are proposed are most convenient for prac- 

 tical application. 



Paired Coefficients — Terminology 



The Fourier integral theorem has been expressed in several slightly 

 different forms to better adapt it for particular applications. It has 

 been recognized, almost from the start, however, that the form which 

 best combines mathematical simplicity and complete generality makes 

 use of the exponential oscillating function g'27r/<_ More recently the 

 overwhelming advantage of using this oscillating function in the 

 discussion of sinusoidal oscillatory systems has been generally recog- 

 nized. It is, therefore, plain that this oscillating function should be 

 adopted as the basic oscillation for both of the proposed tables, A 

 name for this oscillation, associating it with sines and cosines, rather 

 than with the real exponential function, seems desirable. The abbre- 

 viation cis X for (cos x -\- i sin x) suggests that we name this function 

 a cis or a cisoidal oscillation. This term is tentatively employed in 

 this paper. The notation cis(27r//) is also employed where it is 

 desired to use an expression which is essentially one-valued, which 

 avoids the use of exponentials, or which suggests periodic oscillations 

 by its connection with cosine and sine.^ 



^ Since the cisoidal oscillation is simply a uniform rotation at unit distance about 

 the origin in the complex plane, it may be desirable to try some compact notation 

 which directly suggests this rotation; for example, ru(//), W, i*^' might be defined 

 as the complex quantity obtained by rotating unity through // complete turns or 

 4/i quadrants. 



