Exercise · ZW-MOBIUS-TRANSFORMATION-007 · ★★☆ Intermediate
Find the fixed points of \(f(z)=\dfrac{z+2}{z+1}\).
Choose one option.
Show answer
- C. \(\pm\sqrt2\)
Explanation
\(\frac{z+2}{z+1}=z\) gives \(z^2+z=z+2\), that is \(z^2=2\) and \(z=\pm\sqrt2\).
Review the concept: Möbius transformation
A Möbius transformation is a function of the extended complex plane to itself given by a quotient of two linear expressions with complex coefficients \(a,b,c,d\), arranged so that it is not constant.
\[f(z)=\frac{az+b}{cz+d},\qquad ad-bc\neq0\]
More practice problems: möbius transformation
- ZW-MOBIUS-TRANSFORMATION-001 Evaluate f(z)=(z+1)/(z-1) at z=i.
- ZW-MOBIUS-TRANSFORMATION-002 Which of these functions is NOT a Möbius transformation?
- ZW-MOBIUS-TRANSFORMATION-003 If f(z)=(2z+1)/(z+3), what is f(∞)?
- ZW-MOBIUS-TRANSFORMATION-004 Which point does f(z)=(z+2)/(3z-6) send to ∞?
- ZW-MOBIUS-TRANSFORMATION-005 The inverse of f(z)=(2z+1)/(z+1) is f^-1(z)=
- ZW-MOBIUS-TRANSFORMATION-006 If f(z)=1z and g(z)=z+1, what is g(f(z))?
- ZW-MOBIUS-TRANSFORMATION-008 Which Möbius transformation sends 0mapsto1, 1mapsto0 and ∞mapsto∞?
- ZW-MOBIUS-TRANSFORMATION-009 What does the circle |z|=1 become under f(z)=(z+1)/(z-1)?
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