Continued..

besselj

Bessel function of the first kind

Syntax

• J = besselj(nu,Z)
  J = besselj(nu,Z,1)
  [J,ierr] = besselj(nu,Z)
  

Definition

The differential equation

• 

where is a real 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

• 

where is the gamma function.

is a second solution of Bessel's equation that is linearly independent of . It can be computed using bessely.

Description

J = besselj(nu,Z) computes the Bessel function of the first kind, , for each element of the 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.

J = besselj(nu,Z,1) computes besselj(nu,Z).*exp(-abs(imag(Z))).

[J,ierr] = besselj(nu,Z) also returns completion flags in an array the same size as J.

ierr	Description
0	besselj succesfully computed the Bessel function for this element.
1	Illegal arguments.
2	Overflow. Returns Inf.
3	Some loss of accuracy in argument reduction.
4	Unacceptable loss of accuracy, Z or nu too large.
5	No convergence. Returns NaN.

Remarks

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

• 

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

Examples

Example 1.

• format long
  z = (0:0.2:1)';
  
  besselj(1,z)
  
  ans =
                    0
     0.09950083263924
     0.19602657795532
     0.28670098806392
     0.36884204609417
     0.44005058574493
  

Example 2. besselj(3:9,(0:.2:10)') generates the entire table on page 398 of [1] Abramowitz and Stegun, Handbook of Mathematical Functions.

Algorithm

The besselj function uses a Fortran MEX-file to call a library developed by D. E. Amos [3] [4].

besselk

Modified Bessel function of the second kind

Syntax

• K = besselk(nu,Z)
  K = besselk(nu,Z,1)
  [K,ierr] = besselk(...)
  

Definitions

The differential equation

• 

where is a real constant, is called the modified Bessel's equation, and its solutions are known as modified Bessel functions.

A solution of the second kind can be expressed as

• 

where and form a fundamental set of solutions of the modified Bessel's equation for noninteger

• 

and is the gamma function. is independent of .

can be computed using besseli.

Description

K = besselk(nu,Z) computes the modified Bessel function of the second kind, , for each element of the 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.

K = besselk(nu,Z,1) computes besselk(nu,Z).*exp(Z).

[K,ierr] = besselk(...) also returns completion flags in an array the same size as K.

ierr	Description
0	besselk succesfully computed the modified Bessel function for this element.
1	Illegal arguments.
2	Overflow. Returns Inf.
3	Some loss of accuracy in argument reduction.
4	Unacceptable loss of accuracy, Z or nu too large.
5	No convergence. Returns NaN.

Examples

Example 1.

• format long
  z = (0:0.2:1)';
  
  besselk(1,z)
  
  ans =
                  Inf
     4.77597254322047
     2.18435442473269
     1.30283493976350
     0.86178163447218
     0.60190723019723
  

Example 2. besselk(3:9,(0:.2:10)',1) generates part of the table on page 424 of [1] Abramowitz and Stegun, Handbook of Mathematical Functions.

Algorithm

The besselk function uses a Fortran MEX-file to call a library developed by D. E. Amos [3] [4].