110 BELL SYSTEM TECHXICAL JOURNAL 



Campbell and Foster, Fourier Integrals for Practical Application, Bell 

 System Monograph B-584, p. 32 (also Watson, Theory of Bessel Functions, 

 p. 191), we may show that 



iFi\v + -;2v+ 1; -z\ = ^—- Iv{-z/2) (36) 



or setting v = 



^F^{h;l;-z)^e-'''h{z/2) (57) 



which is one of the functions appearing in our work. 



We have also encountered the function iFi (1/2; 2; — s) which is not 

 directly reducible by the above formula. The reduction may be effected 

 in a number of ways. By making use of the relation obtained from (56) 

 by setting v = 1, 



iFi(3/2;3;-s) = "^^ e~"' h{z/2) (58) 



z 



and noting that 



:Fi(l/2; 2; -2) - ,F,(l/2; I; -z) 



^ 1 Y T{m + 1/2) ^ 1_ ^ T{?n + 1/2) ,^ 



r(l/2)^ow!(^ + 1)!^ ^^ r(l/2)^o (m!)2 ^ ^^ 



— 1 '<^ T(m 4- l/2)w , .„ 

 2^ ...w... I tx, (-2) 



r(l/2)iri m\{m + 1)1 ' "^^ (59) 



2 Y^ r(w + 3/2) 



E ^^ ,vr (-^r 



r(l/2) ^0 (m + 2)lm[ 



= |iFi(3/2;3;-2), 



we find that 



iFi (1/2; 2; -s) = e"" [Io{z/2)+h(z/2)] (60) 



It may also be verified by integrating the series directly that 



r iFi(l/2; 1; -z) dz = 2iFi(l/2; 2; -z) (61) 



Jo 



Combining this relation with (57) and (60) above, we deduce the indefinite 

 integrals 



^ The relation (60) was brought to the attention of the author by Mr. R. M. Foster. 



