158 BELL SYSTEM TECHNICAL JOURNAL 



whereas the Fourier integral (2a) gives 



h = - e^' t < 0. 



= / > 0. 



There is another reason why care must be exercised in applying 

 the classical Fourier integral to the Heaviside problem. This is that 

 in solving the operational equation, h = \/H{p), the appropriate 

 expansion of \/H{p) may introduce singularities on or to the right of 

 the imaginary axis in the component terms. This offers no difficulty 

 if either (2) or (3) is employed, but renders the Fourier integral (2a) 

 inapplicable. As an example consider the equation 



V^+ 1 



One form of solution is gotten by multiplying numerator and denomi- 

 nator by V^ — 1 , whence 



Ji = 



_ 4p 



p - 1 p - 1 



and each term has a singularity at ^ = 1. 



A physical interpretation of the foregoing may not be without 

 interest. Suppose that an elementary force F(p)eP^dp, where p = 

 c + io), is applied at an indefinitely remote past (negative) time to a 

 system specified by H{p) . The response of the system is then 



^''^^ e^'dp-^ T^{t)dp, 



H{p) 



where Tp{t)dp is the concomitant transient or characteristic oscillation 

 of the system. If c is chosen sufficiently large then at least for t > 

 the transient term can be made as small as we please compared with 

 the first term. Finally if the impressed force is the unit function 

 (zero before, unity after, time t = 0) and it is written as 



1_ r+*"£!!^ 



5W Jc-i 00 P 



the total response and therefore h{t) is given by 



J_ f 



<^+<«' .tp 



e' 



pllip) 



dp, 



