# 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:

x2y'' + xy' + (x2-n2)y = 0

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.

Bessel functions.

Method

Description

BesselJ0

Bessel function of the first kind of order zero.

BesselJ1

Bessel function of the first kind of order one.

BesselJ

Bessel function of the first kind of integer order n.

BesselJ

Bessel function of the first kind of real order ν.

BesselY0

Bessel function of the second kind of order zero.

BesselY1

Bessel function of the second kind of order one.

BesselY

Bessel function of the second kind of integer order n.

BesselY

Bessel function of the second kind of real order ν.

## Modified Bessel Functions

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

x2y'' + xy' (x2+n2)y = 0

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.

Modified Bessel functions.

Method

Description

BesselI0

Modified Bessel function of the first kind of order zero.

BesselI0Scaled

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

BesselI1

Modified Bessel function of the first kind of order one.

BesselI1Scaled

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

BesselI

Modified Bessel function of the first kind of real order.

BesselIScaled

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

BesselK0

Modified Bessel function of the second kind of order zero.

BesselK0Scaled

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

BesselK1

Modified Bessel function of the second kind of order one.

Table 2. Modified Bessel functions.

## 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.

Table 3. Spherical Bessel functions.

Method

Description

SphericalBesselJ

Spherical Bessel function of the first kind.

SphericalBesselY

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.

Table 4. Airy functions.

Method

Description

AiryAi

Airy function Ai(x).

AiryBi

Airy function Bi(x).

AiryAiPrime

First derivative of the Airy function Ai(x).

AiryBiPrime

First derivative of the Airy function Bi(x).

## 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.

Table 5. Zeroes of Bessel and Airy functions.

Method

Description

BesselJZero

Zero of the Bessel function Jν(x).

BesselYZero

Zero of the Bessel function Yν(x).

AiryAiZero

Zero of Airy function Ai(x).

AiryBiZero

Zero of function Bi(x).