Bessel and Airy Functions

Bessel functions come in many shapes and sizes. Bessel functions arise in many physical problems as the solutions of the following differential equation:

where n is usually an integer. This equation appears when solving certain partial differential equations over a cylindrical domain. The Special class contains static methods for evaluating the Bessel function of the first and second kind of arbitrary integer or real order for real arguments, as listed in the table below.

Modified Bessel functions arise as the solutions of the following differential equation:

where n is usually an integer. The Special class also provides static methods for evaluating standard and scaled versions of the modified Bessel function of the first and second kind of arbitrary integer order for real arguments, as listed in the table below.

Spherical Bessel functions are closely related to ordinary Bessel functions of half-integer order. The Special class also provides static methods for evaluating standard and scaled versions of the modified Bessel function of the first and second kind of arbitrary integer order for real arguments, as listed in the table below.

Closely related to the Bessel functions are the two Airy functions, Ai and Bi. These arise in a number of applications in physics and engineering as two linearly independent solutions of the differential equation

Other solutions to this equation are a linear combination of the two standard Airy functions.

The Special class provides static methods for evaluating the Airy functions and their first derivative. The methods are listed in table 4 below.

The Special class provides static methods for evaluating the zeros of Bessel functions of the first and second kind, and of both Airy functions. For Bessel functions, the order must be supplied as the first argument. The second argument is the index of the zero. For AIry functions, the index of the zero is the only argument. The methods are listed below.