378 ni:i.i. sYsri'.M ri'.ciixic.ii. journai. 



can be represented, however, by the Hniitinsr form assnnied by the 

 series as the InncUunental period 7' is made infmite. That is, we 

 can assume that the function is pericjdic in an inhnite fundamental 

 period and this will not affect the expansion for hnite positi\e and 

 nej^atixe values of time. Proceeding in this wa\' and putting the 

 fundamental period 7" equal to infinity in the limit, the Fourier series 

 (314) becomes an infmite integral and we get 



<^{t)=~ I " n^) ■ cos (col-dio:) ) do: (316) 



H^)= \ I / 0(0 cos a;/ r// +1 / HO s\n o,t dt (317) 



tan 0(a)) = / 4) (t) sin ojI dt^ / (/>(/) cos co/r//. (318) 



,70 ./O 



where 



and 



This is the Fourier integral identity of the function </)(/) and is valid 

 for all finite positive and negative values of time. 



In physical applications, particularly those to electric circuit theory, 

 it is often convenient to employ exponential instead of trigonometric 

 functions. The required transformation follows easily from the 

 relation 



e'^ ^cos d-\-i sin 6, i = \/ —I. 



Thus if we write 2it/T = Wo the Fourier series (312) is easily reduced 

 to the form 



+00 



<^(/) = V F{ino:o)e'"''"' (319) 



where 



F{inu:o) = 1.1 (r) e-'"^"' dr. (320) 



I Jo 



in preciseU' similar manner the Fourier integral (316) can be written 

 as 



4>(t)= I F(icoye''''do: (321) 



= s" / 4>(r)dT I e'^^'-^Ww. (322) 



