GITIDKD-WAVK PKOI'ACATIOX TIIKOrClI C VI{()M ACMOTIC MEDIA 599 



U2 



J3 



Fig. 3 — The function F(x) = 



xJi'(x) 



tion of the infinite product for Ji{x), F(x) is found to be given by 



Fix) = 1 - 2 E -r 



n'ljn^ — X^ 



where the j„'s are the zeros of Ji{x). 



Thus, F(x) is real if x is, which is always the case here. For positive x, 

 F(x) is an always decreasing function of x, which has an infinite number 

 of first order zeros and poles. The zeros are those of J/(.r) and will be 

 denoted by u„ . The poles are the zeros of Ji(x). It may be recalled from 

 the properties of Bessel functions that for large n these zeros and poles 

 are essentially equally spaced with a separation 7r/2. When x is a pure 

 imaginary, equal to jy, F(x) becomes yli(y)/li(y)- This is a steadily in- 

 creasing function of y, always positive, and behaving like y — }-2 for 

 large y. The function F is shown in Fig. 3. Further formulae pertaining 

 to F are given in Appendix I. The inverse function F~ (x), which is also 

 of some importance, is a multivalued function of x, whose behavior is 

 readily understood from the figvire for F(x). We are now ready to proceed 

 with the graphical analysis of the (7-equation. 



In a rectangular coordinate system with X as abscissa and a as ordinate, 

 a contour map is sketched of the function 



