756 Blll.l. SYSTEM TECHNICAL JOURNAL 



Multiplying numerator and denominator of equation (137) by 

 p''\^—\, it Ijecomes 



^-'^-^,^P-^, 038, 



and by direct partial fraction expansion, this is equivalent to 



7_ \^ "ST^ P'm^ _ i "V^ Ptn 



2n+T 2^ ^^"2^+1 ^„ ^^ ^^^^^ 



where 



,■ 2mT 



p„ = e 2-.+! (w/ =0,1,2 ... 2«)- 

 Write, for convenience, 



2n 



and consider the operational equation 



,„= 1 (^VF-/^). (140) 



2n-\-\\p—p,„ p — pm/ 



B>- the rules of the operational calculus, fully discussed in preceding 

 chapters, the solution of this is 



We have now to distinguish two cases: (1) when the real part of Pm 

 is positive, and (2) when the real part is negative. 

 Taking up case (1) first, the preceding can be written 



Repeated integration by parts of the definite integral leads to an 

 asymptotic series, identical with that obtained by applying the Heavi- 

 side Rule to the operational equation (137). 



