146 BELL SYSTEM TECHNICAL JOURNAL 



where the path of integration is from x — i<x) to x -}- i'X) along the 

 line X = const. On account of the analytic nature of the integrand 

 this integral is independent of x (for x > 0). It may be written then 



If / N ,7 1- If 1 rAJ(i\) ,,,,.^, , 



hm :r — : I w(z)e''dz = hm 7—-. I ;r— . I — e'H{tK)dz 



^^0 27r7 J^^^ x^o 27r7 J^^j 27r7 J^ ^X - Z 



= hm T— . T-^ hm + + \^ e~'d{i\)dz 



= lim r^^. (' ;^. r^''4r^^^'^('^)^^2 + <2(^ 5) 1, .r>0, (69) 



where 5 is real and positive. The function Q defined by this equation 

 exists for all values of t and for all values of 5. Similarly, 



lim - — ; I n{z)e^'dz 



= \ lim ;^. f ^. r^''i^e-d{i\)dz + Q{t, 5)1, .v<0, (70) 

 Subtracting and dropping the limit designations 



-^ r w{z)e^^dz - ^. C n{z)e^'dz = ^. C AJ{i\)e^^'d{i\). (71) 



The first integral is zero for / < as can be seen by taking x sufficiently 

 large. Likewise, the second is equal to zero for t > 0. Therefore, 



-U r w(z)e''dz = ^ f AJMe^'^'diiw) = AG{t), <t < ^ (72) 



— - — : I ii(z)e^'dz 



= ^. f AJ{ico)e'"'d(io:) = AG{t) - ^ < / < 0. (73) 

 We may now conclude that 



I n{iy)e'y'diiy) = 0, - co < / < 00 (74) 



provided 



Git) = 0, - ^ < / < 0. (All) 



But (74) is equivalent to 



niz) = 0, (74') 



