238 BELL SYSTEM TECHNICAL JOURNAL 



The hypergeometric function in (28) may always be expressed in 

 terms of K and R by successive applications of recurrence formulae and 

 use of the known relations: 



1; k"" 



1; 



(29) 



By means of the hypergeometric recurrence formuke we may also 

 show that 



. __2l(m—l)k'^-\-n—l^Am-i.n-i-\-im-hn—5)kAm-2,n-2 .,^x 



when })j + w is even. A discussion of the hypergeometric function and 

 a derivation of (30) are given in the appendix. 



From (30), we can compute successively all even order modulation 

 products starting with say Aqo and An known. If negative subscripts 

 occur in applying the formula, they may be replaced by positive sub- 

 scripts without changing the validity of the results; this is proved in 

 the appendix. 



Half Wave Square Law Rectifier — Two Applied Frequencies 

 The solution for two frequencies applied to a square law rectifier, or 

 in fact to any rectifier operating on an integer power law, can be 

 obtained in a manner quite similar to that used in solving the linear 

 rectifier. In the case of a square law rectifier, we have to represent 

 the function 



fix, y) = P^ (cos .T + ^ cos v)2, cos x -\- k cos y > 1 , , 

 = 0, cos X -{- k cos y < O.j 



Going through the same steps with this function that we did with that 

 of (3), we find that the amplitudes of the modulation products can be 

 expressed in terms of K and E as in the case of the linear rectifier; the 

 results are listed in Table I. A set of curves is plotted in Fig. 3. 



We may also show that 



^'+"+1 o^2 00 J"^ ( ttP) Jnl ; irQ ) 



^mn- { ) -^ ^^ (2r - 1)^ ^^'^^ 



when m + w is odd and greater than one and c > \P\ + \Q\. For 



