714 BELL SYSTEM TECHNICAL JOURNAL 



The derivation of the expansion solution from the integral equation 



^.^fme-m 



pmp) 



follows immediateK' from the partial fraction ex[xinsion 



pmp) pii (0) ^ jri^p- pj)piii'ipi) 



where p\, pi ■ . . pn are the roots of the equation Hip) =o, and 



(40) 



//'(/>>)= j//WJ-^^^. (41) 



Partial fraction expansions of this type are fully discussed in treatises 

 on algebra and the calculus and the conditions for their existence 

 established. Before discussing the restrictions imposed on H{p) by 

 this expansion, we shall first, assuming its existence, derive the ex- 

 pansion theorem solution. 



By virtue of (40) the integral c(|uati()ii is 



' 7\ + X i ^ ^^Irrt-H^ = rHOe-^'dl. (42) 



0) ^{p-pi)p}II {P}) ->» 



pm 



The expansion on the left hand side suggests a corresponding expan- 

 sion on the right hand side: that is, we suppose that 



//(/)=/;o(/)+/7,(/)+/;o(^)+ . . .+/,„(0 (43) 



and specify that these compnnein functions shall satisfy the ccjuations 



pm=r'°^'^'~"'' ^''' 



(p-p/pmpr fo°^"^'^'~"" i=i'2--"- (^^) 



It follows at once from (43) and direct addition of equations (44) 

 and (45) that (42) is satisfied and hence is solved provided //,., . . h„ 

 can be evaluated from (44) and (45). 

 Now since 



f e>^'e-'"dl = ~ (46) 



