Bessel Functions of the first kind, denoted as J_n(x), are solutions to Bessel's differential equation. They arise in various physical problems, particularly in cylindrical or spherical coordinates, such as heat conduction and wave propagation. These functions are defined for integer and non-integer orders, and they oscillate between positive and negative values. These functions are important in fields like engineering, physics, and applied mathematics. They exhibit a series of oscillations that decrease in amplitude as the argument x increases. Bessel Functions of the first kind are often used in problems involving circular symmetry, such as vibrations of circular membranes.