DIFFRACTION OF RADIO WAVES BY A I'ARAROIJC CYLINDER 477 



for Vn(z)' The second and third expressions for K„(^) follow from the 

 other asymptotic expressions and (9.11), (9.12). 



Wlien H(n) < 0, the theory of gamma functions and (9.1) lead to 



r(w + l)T{ — n)U„{z) = —T CSC irnUniz) 



r -.-1 , " s (9-1^) 



= / T exp ( — r" — 2r2) (It. 



Jo 



By expressing- \^ir exp [—(z — t)'] as the integral of 

 exp [-T" + 2i(z - t)T] 



taken from t = — oo to + « and substituting in (9.1) it may be shown 

 that, when i2(n) > — 1, 



Un(z) = Fr f e^''--"'e dt, 



J— 00 



Vn{z) = -FT" [ e"^'+'"Vf/r, (9.19) 



Jo 



Wn(z) = -Fi' [ e-'"-''" r" dr, 

 Jo 



F = 2VVr(n + l)7r'''. 



When n is not an integer the path of integration in the integral (9.19) 

 for Uniz) is indented downward at the origin. Equations (9.19) mav 

 also be obtained from (9.1) by using (9.13) and (9.18). 

 When n is an integer 



Un{-Z) = (-)" Un(z), Vn{-Z) = (")" W^iz), (9.20) 



and when ?i is a positive integer 



Uniz) = s,{n,z) = {-y'l:fe-'\ 



n\ dz" 



UUz) = 0, (9.21) 



V.n(z) = -W^niz) = -is2{-n,z) = - -—- ^—^ e\ 

 From Maclaurin's expansion and (9.21), 



E ^"f'n(2) = exp [-r + 2zt]. (9.22) 



