474 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



which will be needed in our work. These functions are related to the 

 parabolic cylinder function Dnizf^'^^ through the equations 



Un{z) = r'' e'"" Dn{2"'z)/T{n + 1), 



Vn{z) = -r" T^^ e''" Z)_„_i {-iz2"')/{2ir)"\ (9.2) 



Wr^iz) = -e 2"^' e'"" D-n-i (iz2'")/(2Trf'\ 



We use the functions Un{z), etc., here instead of Dn(z) because they 

 seem to be more convenient for the particular problem we have to 

 deal with. 



From the definitions (9.1) it follows that Un(z), Vn{z), Wn{z) are one- 

 valued analytic functions of z and n. By expanding exp (2zt) in (9.1) and 

 integrating termwdse it may be shown that 



Uniz) = 2 A COS iwn/2) + izB sin (irn/2), 



Vn(z) = -Ar" -2zi"'^'B, (9.3) 



TF„(2) = -Ai" -2zi''''B, 



When z = and Un(z) = d Un{z)/dz, etc., 



r(l + n/2) ' ' 2r(l + n/2) 



Wn{0) = 



2r(l + n/2) ' 

 fUO) = 2 rinW2) ^,- (0) ^ _^i^ (9.4) 



TFn(O) = 



■(^) 



•8 See E. T. Whittaker and G. N. Watson, Modern Analysis, Fourth Edition 

 (1927) Cambridge Univ. Press pp. 347-354. 



19 W. Magnus and F. Oberhettinger, Formeln und Satze fiir Speziellen Funk- 

 tionen, 2nd Ed., Springer, 1948 Chap. 6 Section 3, and p. 227. A comprehensive 

 account of Dn{z) is given in the forthcoming work, Higher Transcendental Func- 

 tions, compiled by the staff of the Bateman Manuscript Project. 



