1010 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



and interchanging the order of integration of a double integral, 



Similarly, 



so that 



-(X+M+s)«^ 



(s + \)pXs) = 1 + \e-''^''^'>'pXs), 



( N _ s -\- \ -\- y.e , X 



'P^^^^ - (s + x)(s + m) - XMe-'(x+M+«)a ' ^'■''^ 



and 



I I >. — (X+/j+s)5 



^'^^^ (s + \){s + m) - XMe-2^^+''+^^*" 

 Likewise, using (1-4), the Laplace transform /(s) of F{L) is 



1 + X/>i(s) + M?>2(s) 



/(s) = 



A + M + s 



As one might expect from the piecewise analytic character of Pi{L) 

 and Pi{L) there is no convenient way of transforming /(s) back to F{L). 

 By evaluating residues of /(s) exp (sL) at the poles of /(s) one might ex- 

 press F{L) as an infinite series of exponential terms. The most slowly 

 damped term in this series can be expected to approximate F{L) when L 

 is large. The poles of /(s) are at the zeros of the denominator D{s) of ' 

 Pi(s) and p2(s): 



D{s) = (s -F X)(s + m) - \tJ^c-''^-"'^''\ (1-9) 



Since D{x) > for x ^ and both D{ — X) and D{ — ix) are negative, 

 it follows that D{s) has a real zero s = —a ^^■ith a < IVlin (X, ^x). 



The zero s = —a of Z)(s) is the one with the largest real part. For, 

 letting s = .T + iy, we have in the half plane .r ^ —a 



(s + X)(s + m) I - Xm i e" 



-2(\+M+s)« 



^ (.r + X)-(.i- + m) - Xiie"''^^"^''' ^ 0. 



Also, if !j ^ the ^ sign in the above proof can be replaced by > and 

 one concUides that all other zeros of D{s) = satisfy 



Re 6' < - h 



