Find all functions on the Riemann sphere that have a simple pole at \(z=2\) and a double pole at \(z=\infty\), but are analytic elsewhere.
Solution
Goal: Find all functions on the Riemann sphere \(\widehat{\CC}\) that have a simple pole at \(z = 2\) and a double pole at \(z = \infty\), and are analytic elsewhere.
1. A function meromorphic on the sphere with poles only at \(2\) and \(\infty\) is a rational function with those poles.
On the sphere, meromorphic functions are rational (a function holomorphic on the sphere minus finitely many points, with poles there, is a rational function by the theory of meromorphic functions on compact Riemann surfaces / the standard theorem that meromorphic functions on \(\widehat\CC\) are rational).
2. The pole at \(z = 2\) is simple, so \(f(z) = \frac{A}{z - 2} + g(z)\) where \(g\) is holomorphic near \(2\) (indeed \(g\) has no pole at \(2\)).
Laurent expansion at \(z = 2\): principal part is \(A/(z-2)\) with \(A \neq 0\) (simple pole).
3. The only other pole is a double pole at \(\infty\); combined with rationality, \(f(z) = \frac{A}{z-2} + Bz + C\).
A rational function with a simple pole at \(2\) and a pole at \(\infty\) only: the pole at \(\infty\) being double means \(f(z) = P(z) + \frac{A}{z-2}\) where \(P\) is a polynomial.
Since the pole at \(\infty\) is of order exactly 2, \(P\) has degree exactly 2? No — a double pole at \(\infty\) means \(f(1/w)\), as a function of \(w\), has a double pole at \(w = 0\); with \(f(z) = P(z) + A/(z-2)\), the growth at \(\infty\) is governed by \(\deg P\): \(f(z) \sim \text{leading term of } P\). Hence \(\deg P = 2\). Write \(P(z) = bz^2 + cz + d\); subtracting constants doesn’t change poles, so \(f(z) = \frac{A}{z-2} + bz^2 + cz + d\) with \(A \neq 0\), \(b \neq 0\).
4. Q.E.D.
3 gives the general form: \(f(z) = \frac{A}{z-2} + bz^2 + cz + d\), \(A \neq 0\), \(b \neq 0\), with \(c, d \in \CC\) arbitrary.
These are all the functions analytic except for a simple pole at \(2\) and a double pole at \(\infty\).