754 BELL SYSTEM TECHNICAL JOURNAL 



where 4>(t) is determined by the integral equation 



Now from formula (fj of the table of integrals 



2 \ TT./O t\/t 



Vl"' 



whence 



2 \ TT ty/t 



and finally 



1 f' e-'/'" 

 V{1) = -4= / ^—^ dr, where /' = 4//«. 



(129) 



To ron\ert ihis to the form of (127) we write 



F(0 = -4=/ ^</r ~\ ^~j=dr. (130) 



'tt'/o t\ t y/ir'l'' t-\/t 



The value of the intinite integral is known to be unity so that 



1 /•" g-l/r 



(131) 



Now in the integral term of (131) expand e~ '^ in liie usual expo- 

 nential power series and then integrate term by term : the series solu- 

 tion (127) results. This series, while absolutely convergent, is difticull 

 to compute for small values of /; an asymptotic expansion, which can 

 be employed for computation for small values of / is gotten as follows : — 



Write (129) as 



.JIV.»-— i_-m',i,. 



^ T 2\/7r«^o V r 



Repeated partial integrations of this type lead to the series 



''-#-"-i'-(()+-(T)'--!- <'»^' 



It is interesting to note, in passing, that an asymjitotic solution of 

 this type does not appear to be dircclh' dcducible from the operational 

 equation. We observe also that, in this problem, the series in inverse 



