Question
Find the image of the circle under the mapping
. What happens when
?
Answer :
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Let ( w = \dfrac{z - i}{z + i} ). Solving for ( z ) in terms of ( w ), [ w(z+i)=z-i ;\Rightarrow; wz+wi=z-i ;\Rightarrow; z(w-1)=-i(1+w), ] so [ z=\frac{-i(1+w)}{,w-1,}. ] The given circle is ( |z|=r ). Hence [ \left|\frac{-i(1+w)}{w-1}\right|=r. ] Since (|-i|=1),
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