376 BELL SYSTEM TECHNICAL JOURNAL 



by some process of successive approximations. For example, pro- 

 vided the sequence converges, I{t) is the Hmit as n approaches in- 

 finity of the sequence 



Io(t),h{t),I,it), . . . ,/„(/), 



where the successive terms of the sequence are defined by the relations; 

 hit) = Lit) -f^ f'A{t-y)cf>[Io{y)]dy, 



I„+,{t)=Io{t)-fJ'^'A{t-y)<t>[I„{y)]dy. 



We shall not pursue the discussion of non-linear circuits further, in 

 view of their mathematical complexity and their relatively specialized 

 technical interest. The reader who is interested may, however, con- 

 sult the writer's paper on \'ariable Electrical Systems," for a fuller 

 treatment of the subject. 



CHAPTER XI 



The Application of the Fourier Integral to Electric 

 Circuit Theory 



The application of Fourier's series in electrotechnics is a common- 

 place; the use of the Fourier integral, however, has largely remained 

 in the hands of professional mathematicians. An outstanding dis- 

 tinction between the series and the integral, from which the greater 

 power of the latter may be inferred, is that the series represents only 

 a periodic regularly recurrent function, whereas the integral is cap- 

 able of representing a non-periodic function : in fact all types of func- 

 tions, subject to certain mathematical restrictions which are usually 

 satisfied in physical problems. 



Before taking up the application of the Fourier Integral to Electric 

 Circuit Theory, we shall very briefly review the elementary mathe- 

 matics of the series and integral; for a fuller treatment the reader is 

 referred to Byerly, Fourier's Series and Spherical Harmonics. ^^ 



Consider a function (/>(/) , which in the region < / < T is finite, single- 



'i Phys. Rev. Feb., 1921. 



•2 In this chapter the Fourier Integral is approached from the view-point of its 

 physical api^lication and no completeness or rigour is claimed for the treatment. 

 The mathematical theory of the Fourier integral is, of course, completely developed 

 in treatises on the subject. The object of this chapter is merely to outline some of 

 its applications. 



