1. The problem is to understand what a Riemann sphere is. 2. The Riemann sphere is a way to extend the complex plane by adding a point at infinity. 3. It is represented as a sphere where each point on the complex plane corresponds to a point on the sphere, plus one extra point called the "point at infinity". 4. This construction allows us to treat infinity as a regular point, making complex functions easier to analyze, especially for understanding limits and behavior at infinity. 5. Mathematically, the Riemann sphere is the complex projective line \(\mathbb{CP}^1\), which can be visualized by stereographic projection from the sphere to the plane. 6. Important rules: every complex number \(z\) corresponds to a point on the sphere except the north pole, which corresponds to infinity. 7. This concept is fundamental in complex analysis and helps in studying meromorphic functions and conformal mappings.