besselj, bessely (MATLAB Function Reference)

Bessel functions

J = besselj(nu,Z)   Bessel function of the 1st kind
Y = bessely(nu,Z)   Bessel function of the 2nd kind
[J,ierr] = besselj(nu,Z)
[Y,ierr] = bessely(nu,Z)

Definition

The differential equation

where

is a nonnegative constant, is called Bessel's equation, and its solutions are known as Bessel functions.

and

form a fundamental set of solutions of Bessel's equation for noninteger

is defined by:

is a second solution of Bessel's equation--linearly independent of

-- defined by:

Description

J = besselj(nu,Z)

computes Bessel functions of the first kind,

for each element of the complex array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive.

If nu and Z are arrays of the same size, the result is also that size. If either input is a scalar, it is expanded to the other input's size. If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.

Y = bessely(nu,Z)

computes Bessel functions of the second kind,

for real, nonnegative order nu and argument Z.

[J,ierr] = besselj(nu,Z) and [Y,ierr] = bessely(nu,Z)

also return an array of error flags.


`ierr = 1`	Illegal arguments.
`ierr = 2`	Overflow. Return `Inf`.
`ierr = 3`	Some loss of accuracy in argument reduction.
`ierr = 4`	Unacceptable loss of accuracy, `Z` or `nu` too large.
`ierr = 5`	No convergence. Return `NaN`.

Remarks

The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind:

where

is besselj, and

is bessely. The Hankel functions also form a fundamental set of solutions to Bessel's equation (see besselh).

Examples

besselj(3:9,(0:.2:10)') generates the entire table on page 398 of Abramowitz and Stegun, Handbook of Mathematical Functions.

Algorithm

The besselj and bessely functions use a Fortran MEX-file to call a library developed by D. E. Amos [3] [4].