the theory of special functions

Special functions are a collection of mathematical functions that arise naturally when solving complex problems in physics, engineering, and mathematics, especially those involving differential equations. They include functions like sine, exponential, Bessel, and Legendre functions. These functions are "special" because they have well-studied properties, formulas, and applications, making them essential tools for analyzing wave behavior, heat transfer, electromagnetic fields, and quantum mechanics. Instead of trying to solve complicated equations from scratch each time, scientists use these functions because their behavior is well understood, providing a reliable way to describe and predict physical phenomena.