646 BELL SYSTEM TECHNICAL JOURNAL 



nature of coefficient pairs. A beginning has been made by specifying 

 the branches of multiple-valued functions and the method of approach- 

 ing limits. When this has been fully carried out, any pair may be taken 

 from the table and used without the least concern as to the analytical 

 methods by which the validity of the pairs has been established. Thus 

 the finished table will make possible a complete separation of the 

 analytical evaluation of all known Fourier integrals from their practical 

 applications. 



Having now explained, in a general way, the use of Table I, it will 

 be useful to consider in detail a limited number of the pairs which 

 are of special practical interest. 



General Processes for Deriving the Mate 



The table is naturally headed by the two fundamental Fourier 

 integrals (101), (102) because of their intrinsic importance as explicit 

 and implicit definitions of coefficient mates. The chief purpose of 

 the table, however, is to make it possible for the technical man to 

 make the fullest use of coefficient pairs without concerning himself at 

 all as to the analytical work of evaluating either of these Fourier 

 integrals. Pairs (101) and (102) are thus intended to serve mainly 

 as definitions for the pairs which follow. 



The statement has been made that essentially only one Fourier 

 integral has been evaluated by determining the indefinite integral 

 and substituting the integration limits. Whether or not this is pre- 

 cisely true, the statement does illustrate the fact that the formulation 

 of the Fourier integral does not in itself suggest a practical finite ana- 

 lytical process for the actual evaluation of the definite integral. No 

 such system of evaluating definite integrals is known. Writing down 

 the Fourier integral amounts to little more than definitely formulating 

 a question. 



If the coefficient F{f) is expanded as a finite or infinite series in 

 powers of/ (or p), the mate is given by pair (106*), and this involves 

 a finite or infinite series of essentially singular functions which are 

 further considered below in connection with Fig. 3. If a series 

 expansion of F{f) is made in terms of any functions of / for which 

 the mates are known, there is a corresponding series for the mate. 

 Some of these pairs are shown as (104*)-(112). The possibility of the 

 formal infinite expansion does not necessarily imply the convergence 

 of the series in the case of coefficient pairs any more than in other 

 general developments. 



The technical man is not ordinarily a master of infinite series, 

 definite integrals or other infinite processes. It is, therefore, highly 



