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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 Bessel class contains
static methods for evaluating the Bessel function of the first and second kind of arbitrary integer order for real
arguments, as listed in the table below.
| Method |
Description |
| J0(x)
|
Bessel function of the first kind of order zero. |
| J1(x)
|
Bessel function of the first kind of order one. |
| J(n,x)
|
Bessel function of the first kind of integer order n. |
| Y0(x)
|
Bessel function of the second kind of order zero. |
| Y1(x)
|
Bessel function of the second kind of order one. |
| Y(n,x)
|
Bessel function of the second kind of integer order n. |
Table 1. Bessel functions.
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 Bessel
class also contains static methods for evaluating 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 |
| I0(x)
|
Modified Bessel function of the first kind of order zero. |
| I1(x)
|
Modified Bessel function of the first kind of order one. |
| I(n,x)
|
Modified Bessel function of the first kind of integer order n. |
| K0(x)
|
Modified Bessel function of the second kind of order zero. |
| K1(x)
|
Modified Bessel function of the second kind of order one. |
| K(n,x)
|
Modified Bessel function of the second kind of integer order n. |
Table 2. Modified Bessel functions.
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 Airy class contains static methods for
evaluating the Airy functions and their first derivative. The methods are listed in table 2 below.
| Method |
Description |
| Ai(x)
|
Airy function Ai(x). |
| Bi(x)
|
Airy function Bi(x). |
| AiPrime(x)
|
First derivative of the Airy function Ai(x). |
| BiPrime(x)
|
First derivative of the Airy function Bi(x). |
Table 3. Airy functions.
Up: Special Functions Next: Exponential Integrals Previous: Combinatorics and Probability Contents
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