Further Analysis | Part IB, 2002

A function ff is said to be analytic at \infty if there exists a real number r>0r>0 such that ff is analytic for z>r|z|>r and limz0f(1/z)\lim _{z \rightarrow 0} f(1 / z) is finite (i.e. f(1/z)f(1 / z) has a removable singularity at z=0)z=0). ff is said to have a pole at \infty if f(1/z)f(1 / z) has a pole at z=0z=0. Suppose that ff is a meromorphic function on the extended plane C\mathbb{C}_{\infty}, that is, ff is analytic at each point of C\mathbb{C}_{\infty} except for poles.

(a) Show that if ff has a pole at z=z=\infty, then there exists r>0r>0 such that f(z)f(z) has no poles for r<z<r<|z|<\infty.

(b) Show that the number of poles of ff is finite.

(c) By considering the Laurent expansions around the poles show that ff is in fact a rational function, i.e. of the form p/qp / q, where pp and qq are polynomials.

(d) Deduce that the only bijective meromorphic maps of C\mathbb{C}_{\infty} onto itself are the Möbius maps.

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