besselj, bessely (MATLAB Function Reference)

Bessel functions

Syntax

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

is a second solution of Bessel's equation that is linearly independent of and 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 = besselj(nu,Z,1) computes besselj(nu,Z).*exp(-abs(imag(Z))).

Y = bessely(nu,Z,1) computes bessely(nu,Z).*exp(-abs(imag(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.

`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

format long
z = (0:0.2:1)';

besselj(1,z)

ans =
                  0
   0.09950083263924
   0.19602657795532
   0.28670098806392
   0.36884204609417
   0.44005058574493

bessely(1,z)

ans =
               -Inf
  -3.32382498811185
  -1.78087204427005
  -1.26039134717739
  -0.97814417668336
  -0.78121282130029

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

bessely(3:9,(0:.2:10)') generates the entire table on page 399 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].

See Also

airy, besseli, besselk

References

[1] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sections 9.1.1, 9.1.89 and 9.12, formulas 9.1.10 and 9.2.5.

[2] Carrier, Krook, and Pearson, Functions of a Complex Variable: Theory and Technique, Hod Books, 1983, section 5.5.

[3] Amos, D. E., "A Subroutine Package for Bessel Functions of a Complex Argument and Nonnegative Order," Sandia National Laboratory Report, SAND85-1018, May, 1985.

[4] Amos, D. E., "A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order," Trans. Math. Software, 1986.

besseli, besselk beta, betainc, betaln