116 BELL SYSTEM TECHNICAL JOURNAL 



For the same values of x and _v the asymptotic series (4.2-11) gave 



2.40 + 0.171 + .075 + 0.52 + •••• 



If we stop just before the smallest term we get 2.57 for the sum. If we 

 include the smallest term we get 2.65. This agreement indicates that 

 (4.2-11) is actually the asymptotic expansion of (4.2-9). 

 WTien the voltage is of the form 



T' = Q{\ -\- kcos pt) cos qt + Vn 

 we may use 



^ = (2,o)-r(i + |)lf 



(4.2-16) 

 iFir-|;l; -^(1 ^-kco&dAdd 



where R is the envelope with respect to the frequency g/27r and y is given 

 by (4.2-10). The integral may be evaluated by writing iFx as a power 

 series and integrating termwise using the result 



— / (1 + /fe cos ey cos md dd 



(4.2-17) 



where m is a non-negative integer, / any number, 



(a)„, = a(a + 1) • • • (a + m - 1), (a)o = 1, and (0)o = 1. 



The integral may also be evaluated in terms of the associated Legendre 

 function. 



By applying the methods of Section 3.10 to (4.2-16) we are led to 



« " 1 (4.2-18) 



where the as}'mptotic series holds when _\' is ver\' large and k is not too close 

 to unity. These expressions give 



/F/ ~ ^^ {q' f + U2 - (1 - kr"'] + • • •) (-1.2-19) 



