150 BELL SYSTEM TECHNICAL JOURNAL 



The equation holds for all real values of v greater than —1. The symbol 

 F represents the Gaussian hypergeometric function*: 



f (a, 6; .; .) = , + "*. + °(" + D ^^^ + D ,.+ ... (2.2) 



c 1! c(c + 1) 2! 



The derivation of (2.1) requires a rather long succession of substitutions, 

 expansions, and rearrangements, which will be omitted here. 



When V is an integer, the hypergeometric function may be expressed in 

 finite algebraic form, either by performing the integration directly, or by 

 making use of the formulas: 



F{yi/2, — n/2; 1/2; z) — cos (^i arc sin z), 



(2.3) 

 sin (fi arc sin z) 



.(i±-M^-i-.0 



HZ 



together with recurrence formulas for the f'-f unction. When p is an odd 

 multiple of one half, the /-'-function may be expressed in terms of complete 

 elliptic integrals of the first and second kind with modulus [(1 — X)/2] " by 

 means of the relations, 



F(hh;i;k') =-K, 



IT 



F{-h^;^;k') =-E, 



(2.4) 



and the recurrence formulas for the /''-function. For the case of zero bias, 

 we set X = 0, and applv the formula 



F{a. X-a-c; 1/2) = ^J^T]^^M+Zzj\ ^ 



obtaining the result: 



We point out that the above results may be applied not only when the 

 api)lied signal is of the form P cos pt with P and p constants, but to signals 



" For an account of the ])roi)crties of the hypergeometric function, see Ch. XIV of 

 Whittaker and Watson, Modern Analysis, Cambridge, 1940. A discussion of elliptic 

 integrals is given in ("h. XXII of the same hook. 



