PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 659 



to add the Table I pairs together or, in the case of systems 37-39, to 

 apply the two pairs in sequence. In Table II, the customary physical 

 notation is adhered to because it is often of long standing and this 

 necessitates some change in notation when comparing pairs in the 

 two tables. 



Summary and Conclusions 



Many practical applications of the Fourier integral have been 

 simplified by the compilation of Tables I and II, which give coefficient 

 pairs, admittances and transient solutions. 



Minor changes in nomenclature and point of view have been intro- 

 duced, all with the idea of simplifying the practical application of the 

 Fourier integral, in the following ways: 



(1) Using the cisoidal oscillation and the unit impulse side by side 

 as alternative elementary expansion functions. 



(2) Focusing attention upon coefficient pairs for these two ele- 

 mentary functions, both coefficients of a pair representing the resolu- 

 tion of the same arbitrary function, 



(3) Using the frequency and epoch as the parametric variables, in 

 place of the customary radian frequency and independent time 

 variable. 



(4) Employing as a coefficient any real or complex arbitrary func- 

 tion which may be practically useful by regarding it, where necessary, 

 as a limit approached through coefficients which form regular pairs. 



(5) Introducing the ^n(g) functions having an essential oscillating 

 singularity at the origin which mate with ^", the positive integral 

 powers of p. 



(6) Using a notation which greatly reduces the number of occasions 

 for employing the integral symbol in applications of the Fourier 

 theorem. 



Having established the inclusiveness and practical utility of the 

 proposed coefficient pair method of applying the Fourier integral, we 

 are now planning to critically verify the tables and make them as 

 complete as is feasible. It is proposed to include eventually such 

 references to the literature as may add to the interest of the tables. 

 The contributions of integral equations and of the operational method 

 to the present subject will also be incorporated in the tables. The 

 preparation of similar tables for other elementary expansion functions, 

 such as Bessel functions, is also a possibility. A comprehensive table 

 might be made which would include in parallel columns the coefficient 

 functions for a large number of elementary expansion functions, thus 

 giving at once many alternative ways of representing particular time 



