Bessel Functions of the First Kind are a family of solutions to Bessel's differential equation, which commonly arises in problems involving cylindrical symmetry, such as heat conduction and wave propagation. These functions are denoted as J_n(x), where n is the order of the function and x is the variable. They are defined for all real numbers and are particularly important in engineering and physics. These functions exhibit oscillatory behavior and are characterized by their oscillations that decrease in amplitude as x increases. Bessel Functions of the First Kind are used in various applications, including signal processing, vibration analysis, and quantum mechanics, making them essential tools in both theoretical and applied mathematics.