482 thf: bell system technical jourxal, ^l\rcii 1954 



boundaries are given by 



Im U(to) - m] = 0, (10.11) 



or a similar equation involving another pair of saddle points, e.g., ti and 

 ti exp (i2Tr). In this equation z is regarded as fixed and ^o, h are functions 

 of m defined by (10.2). It should be noted that although (10.8) defines 

 a path of steepest descent in the ^-plane, (10.11) defines curves (bound- 

 aries of regions) in the ?/? -plane. 



5. If m is such that the path of integration for a particular function, 

 say Un{z), passes through both to and ^i, each one will contribute to 

 the value of Un{z). Furthermore, if m is such that 



Reim -m] =0, (10.12) 



U and t\ have the same height and the two contributions have a chance 

 of cancelling each other and giving a value of zero for Vniz). Thus 

 (10.12) or some similar equation defines the lines in the »i-plane along 

 which the zeros of Uniz), etc., (regarded as functions of m) are asympto- 

 tically distributed. 



6. The lines in the w -plane defined by (10.11) and (10.12) may be 

 obtained by substituting the values (10.2) for U and h in 



m) - Kh) = h' - t{ - 2Wi log (/o//i), (10.13) 



and setting the imaginary and real parts, respectively, to zero. How- 

 ever, instead of dealing with m directly it is easier to use w = u -\- iv \ 

 defined by 



w = log {h/h) = log I k/ty I -f tXarg h - arg ^i), (10.14) | 

 m = ^'/(cosh w + 1), (10.15) | 



where (10.15) follows from (10.14) and (10.2). Then (10.13) becomes 

 /(/o) - .f{h) = /«(sinh w - w) 



^ z'ismh w - w) (10-16) 



cosh w -\- 1 



The inequalities (10.7) show that 



M ^ 0, I V I ^ TT. 



7. For the special case z = p exp (?7r/4), (10. IG) gives 



(cosh u + cos V — V sin v) smh w = (cosh u cos v + 1) u, (10.17) 

 (cos V + cosh u -{- u sinh u) sin v — (cosh u cos v + 1) v, 



