root E can be derived by making a change of variables to transform E 

 into Y > as was done in $9. 



C. Incompressible Flow Problems 



Incompressible flow results follow from taking the limit k ■*■ 0, 



o 



which has the unfortunate effect of reducing the strip of analyticity D 



to a line on which the functions are continuous, but not necessarily 



analytic. The branch cuts from ± k also join up to form a complete 



barrier along, say, the imaginary axis. To avoid possible difficulties 



stemming from this it is usual to work with finite k and then let k ■* 0. 



o o 



This, however, makes for unnecessary complications in much of the wcrk, 



and it is useful to be able to tackle the incompressible problem directly. 



To this end we imagine ,the branch cuts as starting from + iO and going 



to infinity in Im s > and from 0-i0 to infinity in Im s < 0. 



Then the limiting form of the function y is real and positive 



s 



on the whole real s-axis, i.e., it is there the function |s|. We shall 

 l 



write (s 2 ) for this function in the complex plane with the cuts as 

 indicated; it is the continuation to complex s of the function |s| on the 

 real axis, and can also be defined as s(sgn Rea) where sgn x ■ ± 1 for 



x-< 0. Thus 



l 

 Y ■* (s 2 ) T " s (sgn Res) as k + (10.21) 



The multiplicative split is 

 l i i 

 (s 2 ) T - sT s T (10.22) 



li l 



where s£ means the branch of s 2 " which behaves like s 2 " as s •*• +«>with a 



li 

 cut from - 01 in the lower half -plane, while s" 5 " behaves like s 2 " as 



s -*■ + « but has the cut from 0+0i in the upper half plane. 



79 



