modified Bessel functions

Modified Bessel functions are special mathematical functions that solve certain types of differential equations, often appearing in physics and engineering problems involving heat transfer, wave propagation, or circular symmetry. They are similar to standard Bessel functions but are adapted for situations where the solutions grow or decay exponentially rather than oscillate. These functions help describe how quantities like temperature or electromagnetic fields behave in complex systems, especially when symmetry simplifies the problem. In essence, modified Bessel functions are essential tools for modeling and understanding phenomena with radial or cylindrical features where exponential changes occur.