520 BELL SYSTEM TECHNICAL JOURNAL 



tion Y in the form 



F(to. + io^) = Yiic,.) + ^ F(i)(ico.) + ^' Y^'Ki^c) + • • • 



Substitution of (26) in (25) gives 



/ = E-exp (icoj + id) fl—.-T-^ Y{i(^c) f* co"F(^'a))e»'<-'(ia;. (27) 

 n.dcxjc" J_^ 



But by the identity (24) and repeated differentiations with respect 

 to /, we have 



I o}F(io})e''''dcc = n exp (i I ndt) , 



r i^''F{ii^)e"''d^ = (n^ - i^\exp(i f'fxdi), 



• • (28) 



I w"F(ico)e'"^doi = Cn exp lij fxdtj • 



Substitution of (28) in (27) gives 

 / = £ exp ( i r mt -{-ioYl F(ico,) + f 1 C„ ^ F(^-co,) | , (29) 



which agrees with (16). 



Formula (25), as it stands, includes the initial transients at time 

 / = as well as any which occur at discontinuities in /^(^). Differ- 

 entiation with respect to / under the integral sign, however, in effect 

 eliminates these transients and (29) leaves only the quasi-stationary 

 current (plus the correction series given in (19)). 



The series appearing in formula (29) may not be convergent; in any 

 case its computation is laborious. Furthermore, in its application to 

 the theory of frequency modulation, terms beyond the first two 

 represent distortion. For these reasons it is often preferable to 

 proceed as follows: 



Returning to formula (25), we write 



Yd.. + i„) = ( 1 +^,^^ + . . . +^;£-) F(.-..) + i?.(»„ .), 



(30) 



