Complex Methods | Part IB, 2001

Consider a conformal mapping of the form

f(z)=a+bzc+dz,zCf(z)=\frac{a+b z}{c+d z}, \quad z \in \mathbb{C}

where a,b,c,dCa, b, c, d \in \mathbb{C}, and adbca d \neq b c. You may assume b0b \neq 0. Show that any such f(z)f(z) which maps the unit circle onto itself is necessarily of the form

f(z)=eiψa+z1+aˉz.f(z)=e^{i \psi} \frac{a+z}{1+\bar{a} z} .

[Hint: Show that it is always possible to choose b=1b=1.]

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