Exercise · ZW-MOBIUS-TRANSFORMATION-010 · ★★★ Advanced
Let \(f(z)=\dfrac{z+1}{z-1}\) and \(g(z)=\dfrac1z\). Using the product of coefficient matrices, \(f(g(z))=\)
Choose one option.
Show answer
- A. \(\dfrac{z+1}{1-z}\)
Explanation
With \(F=\begin{pmatrix}1&1\\1&-1\end{pmatrix}\) and \(G=\begin{pmatrix}0&1\\1&0\end{pmatrix}\), the product is \(FG=\begin{pmatrix}1&1\\-1&1\end{pmatrix}\), which corresponds to \(\frac{z+1}{-z+1}\). Order matters: \(GF\) gives \(g(f(z))=\frac{z-1}{z+1}\).
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∞?