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Unformatted text preview: Extra Problems Sheet 3 Stephen Taylor May 4, 2005 1. Define f : C ∪ {∞} → C ∪ {∞} by f ( z ) = az + b cz + d where a, b, c, d are fixed complex numbers. Show that f is onetoone and onto if ad bc 6 = 0 . What happens if ad bc = 0? One to one : Suppose f ( z ) = f ( w ) where w 6 = z . Then az + b cz + d = aw + b cw + d ⇒ ( az + b )( cw + d ) = ( aw + b )( cz + d ) ⇒ azd adw = bcz bcw ⇒ ad ( z w ) = bc ( z w ) ⇒ ( z w )( ad bc ) = 0 So if ad bd 6 = 0 then z = w , which shows that f is onetoone. Consider the case f ( z ) = ∞ f ( w ). Then cz + d = 0 = cw + d . c 6 = 0, otherwise ad bc = 0. So z = w , and we have shown then function is onetoone. Onto : Let w = f ( z ) ∈ C ∪ {∞} . Then w = az + b cz + d ⇒ wcz + wd = az + b ⇒ z ( wc a ) = b wd ⇒ z = b wd wc a So the above shows that f is onto everywhere except in the case where w = a/c . Consider the case f ( z ) = a/c . This happens iff z = ∞ which shows the function is onto everywhere. 2. Show that if z, w ∈ C ∪ {∞} , then d ( z, w ) = d 1 z , 1 w We note the metric on C ∪ {∞} is defined to be 1 d ( z, w ) = 2  z w  p (  z  2 + 1)(  w  2 + 1) So by definition d 1 z , 1 w = 2  z 1 w 1  p (  z 1  2 + 1)(  w 1  2 + 1) = 2  zw  z 1 w 1 ...
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Math, Complex Numbers

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