
#1
Aug1310, 11:05 AM

P: 76

1. The problem statement, all variables and given/known data
Mobius Transformation copies the annulus [tex] \{ z: r<z<1 \}[/tex] to the region bounded by the discs [tex] \{ z : z\frac{1}{4} = \frac{1}{4} \} [/tex] and [tex]\{ z: z=1 \}[/tex] . Find r Hope you guys will be able to help me! Thanks a lot! 2. Relevant equations 3. The attempt at a solution Got no idea...Hope you'll be able to help 



#2
Aug1310, 03:05 PM

P: 360

There seems to be an error in the definition of the annulus, since the lower and upper bounds for z should be different.




#3
Aug1310, 05:28 PM

P: 76

You're right... I'm sry... I've corrected my typo




#4
Aug1410, 03:40 PM

P: 360

Mobius TransformationsComplex Analysis
OK, here are some hints:
Do you know, or can you prove, that if z and [itex]\alpha[/itex] are complex numbers such that [itex]\overline{\alpha}z \neq 1[/itex]. then [tex]\left  \frac{z  \alpha}{1  \overline{\alpha}z} \right  < 1,[/tex] if z < 1 and [itex]\alpha[/itex] < 1, and [tex]\left  \frac{z  \alpha}{1  \overline{\alpha}z} \right  = 1,[/tex] if z = 1 or [itex]\alpha[/itex] = 1. From this, it follows that the transformation [tex]T(z) = \frac{z  \alpha}{1  \overline{\alpha}z}[/tex] maps the unit disk to the unit disk. Now find a value of [itex]\alpha[/itex] such that T maps {z: z < r} to {z: z  (1/4) < 1/4}. That should do it. Please post again if you have any other questions. Petek 


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