Complex analysis question I don't understand solution

[Genericcoder]

So I am self studying complex analysis using some notes online so here he is trying to figure the map of certain domain applied to a function I follow everything but the end I don't get in the end when he got Re(w) < 1 how did he get that ? is it just because Re(w) which is 1/2 and 0 in this case is always less than 1? or what I don't get it.

f(S) = {w ∈ C : |w −1/2| > 1/2, Re(w) < 1}.

 

[PeroK]

Deleted.

 

[Mentallic]

Notice at the beginning of the 3rd page, the first criteria given is u<0 which was kind of neglected until the end.

 

[Genericcoder]

but why exactly 1 ? I don't see how would that make it the case... ? @ Mentallic

Perok ?

 

[Mentallic]

but why exactly 1 ? I don't see how would that make it the case... ? @ Mentallic

Because u<0 which is Re(1/w)<0 was one of the criteria we found that needs to be true after we did the inversion z->1/w, but then we needed to do an expansion z->2z, and u<0 simply becomes u<2(0) which is still u<0, so nothing changes in that step while the circle did change. The final step is a translation of z->z+1 so if u<0 then this becomes u<0+1=1. Hence, Re(w)<1.

 

[Genericcoder]

Thank you makes sense.