ELECTRIC CIRCUIT THEORY 2,77 



valued aiul has oiih' a finile nuinbcr of discontinuities or of maxima or 

 minima. In this rcs^ion il can then he expressed as the I'^iurier series 



where 



<^(/) =\Ao+^\An COS (^J!' / ) +Bn siu (^^" /, ) | (312) 



1 •- ' 



An = l f%{t)- COS {^^"11! l)dt, 



« = f. J'% (/)• sin (^^"" /).//. 



(313) 



B 



An equivalent series is 



0(O = i Fo+^FnCos (^p/-0„) (314) 



1 

 where 



Fn = \^Al^Bl, 

 e„ = tan-i(-5„M„)- 



(315) 



This expansion is valid in the region 0<t<T, irrespective of the 

 form of the function elsewhere. Let us, however, assume that the 

 function repeats itself in the period T: that is 



4>{t±kT)=^4>{t), k = l,2,3 . . . N. 



Then the expansion represents the function in the region — AT < t < NT. 

 Finally if A" is made infinite, the function is truly periodic and the 

 Fourier series represents it for all positive and negative values of 

 time. 



It follows from the foregoing that, if the Fourier series represents 

 the function for all positive and negative values of time, the function 

 must be periodic for all positive and negative values of time; otherwise 

 the expansion is valid only over a restricted range of time. 



Now let us suppose that 0(0 is non-periodic. For convenience, 

 in connection with subsequent applications we shall suppose that it is 

 zero for all finite negative values of time, that it converges to zero as 

 /— ^00 , and that 



£ 



4>(l) (It 



exists. Such a fund ion obxiously cannot be represented by the usual 

 Fourier series for all finite positive and negative values of time; it 



