An explicit expression for the square root occurring in Equations 

 [31a,b] can be found from the general formula: 



(a + ib)^/^ = ± (^ + 1^) ; u = |/| (a + /? + b^] [32a,b] 



Here a and b are any two real numbers (except b = ifa<^0) and y 

 denotes, as usual, the positive square root of a positive real niamber. If 



a <_ and b = 0, (a + ib) =+_iV|a|. It will be convenient to denote 



1/2 

 by (a+ib) the value given by [32a] with use of the plus sign. 



[Note that (a+ib) (a+ib) = a + ib.] (a+ib) is a continuous 



+ + + 



function of a and b except for a discontinuous jump as b varies past zero 

 with a < 0, the jump as b rises from negative to positive being readily 

 found to be from -i Vja\ to i VT^I • [To verify [32a], solve [32b] for b 

 and substitute in [32a] squared.] 



Putting a = e and b = - g into Equations [32a,b] gives: 



(e - ig)^' = u - IJ, u=\/ i(e+V?TT) [33a,b] 



Since for a damped beajn g > , no difficulty can occur if e < (which is 

 qiiite unlikely). 



Also, from Equations [32a,b] with a = e + ^ (e - g^ ) , 



2 



b=-g(l+2^e), the square root indicated in Equations [31a,b] has the 



value 



where 



[e - ig + 5^ (e - ig)^]^^^ = w - ^ (1 + 2Ce) [3Ua] 



+ 2w 



w =/|(a+/Z7^) [3k^: 



32 



