The function J_n(x) , known as the Bessel function of the first kind, is a solution to Bessel's differential equation. It is commonly used in various fields such as physics and engineering, particularly in problems involving cylindrical symmetry. The subscript n represents the order of the function, while x is the variable. Bessel functions, including J_n(x) , appear in applications like wave propagation, heat conduction, and vibrations of circular membranes. They are characterized by oscillatory behavior and are defined for both positive and negative values of x . Their properties make them essential in mathematical modeling of physical phenomena.