ReyChiquito
Sep22-04, 12:42 AM
Hello, i have this q.
The problem is that i need to transform the region 0<Re(z)<pi/2 into the unit circle. Now here is what ive done.
First i transform z_1=iz (rotate pi/2)
then z_2=Im(z_1)/(pi/2-Im(z_1)) (expand the band onto the whole upper plane)
and then z_3=(1/2+iz_2)/(1/2-iz_2) (transform the upper plain into the unit circle).
Now, substituting i get w(z)=(pi/2-Re(z)(1-2i))/(pi/2-Re(z)(1+2i)).
Is this correct?
Ive shown that the transformation is 1-1. Is this enough to state that is conformal or do i need to prove something else (that it is analytical, wich is not clear for me). If so, what criteria would you suggest (less work)?
Plus, i need to do the same with the intersection of two discs (r=1 centers (1,0) and (0,1)) and in that case i have no clue how... should i send one part of the boundary to y=0 and the other to y=infinitum and then do the last transform or there is an easier way?
ps. sorry for bad english
The problem is that i need to transform the region 0<Re(z)<pi/2 into the unit circle. Now here is what ive done.
First i transform z_1=iz (rotate pi/2)
then z_2=Im(z_1)/(pi/2-Im(z_1)) (expand the band onto the whole upper plane)
and then z_3=(1/2+iz_2)/(1/2-iz_2) (transform the upper plain into the unit circle).
Now, substituting i get w(z)=(pi/2-Re(z)(1-2i))/(pi/2-Re(z)(1+2i)).
Is this correct?
Ive shown that the transformation is 1-1. Is this enough to state that is conformal or do i need to prove something else (that it is analytical, wich is not clear for me). If so, what criteria would you suggest (less work)?
Plus, i need to do the same with the intersection of two discs (r=1 centers (1,0) and (0,1)) and in that case i have no clue how... should i send one part of the boundary to y=0 and the other to y=infinitum and then do the last transform or there is an easier way?
ps. sorry for bad english