The function I_n(x) represents the modified Bessel function of the first kind of order n . It is commonly used in various fields such as physics and engineering, particularly in problems involving cylindrical symmetry. The function is defined for real or complex arguments and is often expressed as a power series. Modified Bessel functions, including I_n(x) , arise in solutions to differential equations, especially in scenarios involving heat conduction and wave propagation. They are characterized by their oscillatory behavior and exponential growth for large values of x . These functions are essential in mathematical modeling and analysis.