Associated Legendre functions are crucial in solving partial differential equations, particularly in contexts involving spherical coordinates and the Riemann sphere. They are commonly applied in physics, especially in the analysis of electric and magnetic field evolution. The discussion highlights their utility in visualizing certain physical phenomena, although not all PDEs can be represented in this framework. Additionally, there is curiosity about the integration of Associated Legendre functions with other functions over the real line, suggesting a need for clarity on their broader applications. Overall, these functions play a significant role in mathematical physics and complex analysis.