96 BIRKHOFF AND LANGER. 



and it is readily verified that there corresponds to each choice of an 

 arbitrarily small positive constant x» some value of .r, say x = X , 

 such that 



(72) 



(i) |1 -f(-z)\ < X , 



(ii) I c 3 ~ ~U^u I < X, for x g X. 



We shall assume that x is chosen sufficiently small to preclude the 

 vanishing of /(z) outside or on the boundary of the region | x | ^ X. 



Now /(z) is analytic throughout the entire finite plane, and hence 

 it is possible to find in any interval of the Y axis, however small, some 

 point y = i/o which is such that the line y = y contains no zero of /(z). 

 Let y = \\ and y = Y% be any two such lines and consider the rect- 

 angle K bounded by them and the lines x = X and x = — X. We 

 shall determine the number of zeros of/(z) within K by observing the 

 increase in arg/(z) as z describes the perimeter. 



We have arg/(z) = sin -1 , where l\f(z)\ denotes the coeffi- 



I /(z) I 



cient of V — 1 in the expression for f(z). Moreover, for y = constant 



I {/(z) } has the form 



Hf(z)} =d 1 e x + d,e rx + d 3 el^ x , 



where the coefficients, (/,, i — 1, 2, 3 are real constants. The finite 



zeros of /{/(z)}, and hence those of , being the roots of the 



I /(z) I 

 equation 



di + foW' + d*" = 0, 



are, however, separated by the finite zeros of the derivative of the left- 

 hand member, namely by the roots of the equation 



f/ 2 {r- 1} +d 3 rc x = 0. 



Since this equation is satisfied by at most one value of x it follows that 



^— vanishes at most twice, and consequently that arg /(z) changes 



! /(z) I 



