Exercise · ZW-MOBIUS-TRANSFORMATION-009 · ★★★ Advanced
What does the circle \(|z|=1\) become under \(f(z)=\dfrac{z+1}{z-1}\)?
Choose one option.
Show answer
- B. The imaginary axis
Explanation
The point \(z=1\) of the circle goes to \(\infty\), so the image is a line. For \(z=e^{it}\), \(f=\frac{e^{it}+1}{e^{it}-1}=-i\cot\frac t2\), purely imaginary.
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-007 Find the fixed points of f(z)=(z+2)/(z+1).
- ZW-MOBIUS-TRANSFORMATION-008 Which Möbius transformation sends 0mapsto1, 1mapsto0 and ∞mapsto∞?
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