490 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



In Fig. 11.4 the dashes mean, as before, that the path of steepest descent 

 is on a sheet of the Riemami surface other than | arg t | < ir. The alter- 

 nate dots and dashes are used to indicate that | arg t | > tt and that in 

 addition the path lies directly under the curve it parallels. Although in 

 Fig. 11.4 the two kinds of dashed curves are joined at about arg / = 

 — Sir — 7r/4, they actually should spiral in to / = before they connect. 



8. I m I = 0.005p", arg m = x. Fig. 11.5. For arg w = tt, (11.6) shows 

 that ^1 and ti exp ( — 2xi) are of the same height. 



9. m = 0.005p", arg m = 37r/2. For tt < arg m < Sir/2, li exp ( — 27rz) 

 is higher than ^i and the paths spiral into t = comiterclockwise. At 

 arg m = 37r/2 the rate of spiralling is zero and we ha\'e the path shown 

 in Fig. 11.5 (which is the path for arg m = x/2 rotated by 180 degrees). 

 Here arg h = 57r/4. 



10. m = 0.005p", arg m = — 7r/2. The paths for arg ?n equal to — 7r/2 



Uf 



>t, 



^\ t-PLANE 



Fig. 11.4 — Deformation of path of integration U into path of steepest descent 

 through ti when m = 0.005 p^ e.xp (iV/2 + iO.Ob). 



