07/04/2022, 11:37 PM
Hey, tommy.
So a transcendental entire function does not have a fixed point at infinity you are right. Due to Picard's theorem.
But a transcendental entire function restricted to \(\Im(z) > 0\), can absolutely have a fixedpoint at \(i\infty\). We usually just call that infinity though, where from context it's implied that it's \(i\infty\) because we're in the upper half plane.
This is no different than saying that \(e^z\) has a zero at \(\infty\) when \(\Re(z) < 0\). Kind of an abuse of notation, but not really when you think of \(\infty\) as a boundary value of a simply connected domain on the Riemann sphere.
So a transcendental entire function does not have a fixed point at infinity you are right. Due to Picard's theorem.
But a transcendental entire function restricted to \(\Im(z) > 0\), can absolutely have a fixedpoint at \(i\infty\). We usually just call that infinity though, where from context it's implied that it's \(i\infty\) because we're in the upper half plane.
This is no different than saying that \(e^z\) has a zero at \(\infty\) when \(\Re(z) < 0\). Kind of an abuse of notation, but not really when you think of \(\infty\) as a boundary value of a simply connected domain on the Riemann sphere.
