Urn V B+h) " V 9) 

 Y'(8) - *** -i - (2.6) 



+ h ■*■ o n 



as the point s+h approaches s along any path in the plane. 



In a precisely similar way, the half-range transform of the scattered 

 displacement y(x) for x < 



Y_(s) - fy(x) exp isx dx (2.7) 



is an analytic function of s in a lower half of the s-plane, 



R : Im s < + k . (2.8) 



oi 



and for a function y(x) integrable at x - 0- Y_(e) has non-growing 



algebraic behavior at infinity in R , so that |Y_(s)| » 0(|s|~ u ) for 



some (j > as |s| ■*■<*> along any radius in R . (Algebraic growth of 



Y (s) is permitted if y(_t) has a non-integrable singularity as x ■+ 0-.) 



Since k J and k are strictly positive, the FULL- RANGE FOURIER 

 ol U 



TRANSFORM 



Y(s) » Y + (s) + Y_(s) - y(x) exp isx dx (2.9) 



_CO 



exists as an analytic function of s in a strip D, 



D'EJR :-l..< Ias<U J " ' (2.10) 



+ - '1 oi 



formed by the intersection of the overlapping upper and lower half-planes 

 R and R_. [See Figure 2.] 



The notation and conventions for the Fourier transforms are such 

 that©f unctions ariae from half-range transforms over the positive 

 x-axis, Q functions from half-range transforms over the negative x-axls. 



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