# Bessel and Airy Functions

## Bessel functions

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

*x*

^{2}y'' + xy' + (x^{2}-n^{2})y = 0where *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.

Method | Description |
---|---|

Bessel function of the first kind of order zero. | |

Bessel function of the first kind of order one. | |

Bessel function of the first kind of integer order | |

Bessel function of the first kind of real order | |

Bessel function of the second kind of order zero. | |

Bessel function of the second kind of order one. | |

Bessel function of the second kind of integer order | |

Bessel function of the second kind of real order |

## Modified Bessel Functions

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

*x*= 0

^{2}y'' + xy' (x^{2}+n^{2})ywhere *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.

Method | Description |
---|---|

Modified Bessel function of the first kind of order zero. | |

Scaled modified Bessel function of the first kind of order zero. | |

Modified Bessel function of the first kind of order one. | |

Scaled modified Bessel function of the first kind of order one. | |

Modified Bessel function of the first kind of real order. | |

Scaled modified Bessel function of the first kind of real order. | |

Modified Bessel function of the second kind of order zero. | |

Scaled modified Bessel function of the second kind of order zero. | |

Modified Bessel function of the second kind of order one. |

## Spherical Bessel Functions

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.

Method | Description |
---|---|

Spherical Bessel function of the first kind. | |

Spherical Bessel function of the second kind. |

## Airy Functions

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

*y'' - xy*= 0.

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.

Method | Description |
---|---|

Airy function Ai( | |

Airy function Bi( | |

First derivative of the Airy function Ai( | |

First derivative of the Airy function Bi( |

## Zeros of Bessel and Airy Functions

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.

Method | Description |
---|---|

Zero of the Bessel function | |

Zero of the Bessel function | |

Zero of Airy function Ai( | |

Zero of function Bi( |