726z - b = r2ei62,A= IAleilT
Then!( )
z -_ vi Ah r2 e (1/2)i(^111 +o^2 -2w+1!"+2k1r) ,
rfrom which we select
fo(z) = vJAhr2 e(l/2)i(o 1 +o 2 -2w+1l")
r= i vJAhr2 e(1/2)i(o 1 +o 2 -2w)
rChapter9k = 0,1resulting in a distribution of values of fo(z) along the x =axis as follows:
For 0 < z = x < a we find thatJO^1 ( ) _ X - -z. ./JAhr2
rFor z x > b,
For a < z = x < b, and x on the upper boundary of the cut,fo(x) = i vlAlr1r2 ei1r/2 = _ vlAhr2
r r
For a < z = x < b, and x on the lower boundary,fo(x) = i vlAhr2 ei1r/2 = vlAJr1r2
r rNext, consider a circler+: z = Rei^6 , 0:::; (J:::; 271", with R > b, a small
circle 1i: z = rei^6 , 0 :::; (J :::; 271", with 0 < r < a, and a contour >. consisting
of the lower boundary of the cut from a+E to b-E [O < E < min(a-r, R-b)]
followed by the circle 1t= z - b = Eei^6 , -71" :::; (J :::; 7r, then the upper
boundary from b-E to a+E, and then the circle 1i: z-a = Eei^6 , 0:::; (J :::; 271",
as shown in Fig. 9.28. Note that E is to be taken small enough so thata+ E < b - E.
Again, the strong form of the Cauchy-Goursat theorem gives