I'LOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1195 



X < 0. The single- valued monotone functions Ai , A2 , and A3 are defined 

 respectively by the restrictions A > 1, 1 > A > 0, and A < 0. When A is 

 used without subscript it is implied that either Ai , A2 , or A3 can be used. 

 It will be useful to remember that 



A'W = ^^%:fi. (88) 



A{x) 



In terms of the function A, (85) integrates to 



<U(3C,/)=bW-.lA[|)^] (.^^^ (89a) 



(m(t) : arbitrary function) 



or 



ai(3C, /) = m{t) - 3C (j = 7). (89b) 



The latter case (j = 7) corresponds to T) = V{t) and hence was included 

 in Section G. Therefore in the following we shall consider only j 9^ 7. 

 Now by making use of (89a) and (88), (86) can be rewritten in the form: 



eN ^-j + y'^ {j - y){^ -j + y) dt j-y "^ 



f primes denoting here — j . 



We now observe that the right side of (90) is harmonic, while tKe left side 

 is a function only of JC and /. From this it follows that the right side can be 

 written in the form: 



aoc 



dt 



/-(^ - OC) _ ^, ^ (^) r.«^^1 ^ ^(^) (91) 



J-y Lj -y A 



From (90), (91) and (89a) follows 



?^(ft(cu) = -^ni 



J-y (92) 



01 \ J - y \j - y !/ 



Since (92) is of the form 



The result of taking f — j of the right side must be zero identi- 



^^^^^ = const 



