DIFFRAT'TIOX OF RADIO WAVKS ^\\ A l'\i;\HOI,I(' CYLINDKH 499 



Table 12.4 — Leading Terms in the Asymi'totic Expansions for 



Un(z), Vn(z), Wn{z) WHEN Z = iT^'-p AND I 2/1 I » p' 



The notation used in Table 12.4 is as follows: 



m = n -\- I, i = exp (i7r/2), —3ir/2 ^ arg in < 7r/2, 



nr n-zn —1/2 



B = 2 IT exp 



m 



1 - 1 



og 



2 V °2 



a'a = exp [ — p(2im)^''"^], 

 a[ = exp [p{2imf''^ = l/a'o, 



(12.11) 



B' may be expressed in terms of gamma functions with the help of (12.10.). 

 Approximate expressions for the zeros are given by the complex 

 conjugates of (12.7). For example, if fc be a large positive integer such 

 that 2/v » p", the zeros of Wn{i~^'~p) are at n = n(k) where i^"ai — exp 

 (iTk) and 



}i(k) 



■2k -\- i ■'4p/i-'''/7r — 47pV7r'. 



(12.12) 



Here the approximation has been carried out one step further than in 

 (12.7) We also have for the quantities in (7.11) 



[aiF„(r^>)/a«] „=,.(.) ^ (-Y^'B'i[7r - p(i/ky% (12.13) 



[i""F„(r^^-p)/sin xn] „=„(,) ^ (-y-''2B'i. 



13. asymptotic expressions for r„(2), ETC., WHEN fl IS NEAR Z-/2 



The asymptotic expressions given in Section 12 fail when n is near 

 z /2. Expressions for the parabolic cylinder functions which hold for 

 this region have been given by Schwid.* More recent studies of this sort, 

 based on differential equations, have been made by T. M. Cherry" and 

 F. Tricomi.^" Their results suggest the possibility that our expressions 



* Reference 20, page 478. 



^* Uniform Asymptotic Expansions, J. Lond. .Math. Soc, 24, pp. 121-130, 1949. 

 Uniform Asymptotic Formulae for Fund ion.s with Transition Points, Am. Math. 

 Soc. Trans.; 68, i)p. 224-257, 1950. 



" Equazioni Differenziali, Einaudi, Torino, pp. 301 308, 1948. 



