bessely - Bessel function of second kind

Note   In a future release, the behavior of bessely will change in the following ways: The syntax [Y,ierr] = bessely(...) will be removed. bessely will not accept nu and Z passed as a combination of row and column vectors. bessely will not accept nu and Z passed as a combination of non-scalar and empty inputs.

Syntax

Definitions

The differential equation

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

A solution Y_ν(z) of the second kind can be expressed as

where J_ν(z) and J_–ν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν

and Γ(a) is the gamma function. Y_ν(z) is linearly independent of J_ν(z).

J_ν(z) can be computed using besselj.

Description

Y = bessely(nu,Z) computes Bessel functions of the second kind, Y_ν(z), 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.

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

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

ierr	Description
0	bessely successfully 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.

Tips

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

where is besselh, J_ν(z) is besselj, and Y_ν(z) 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)';

bessely(1,z)

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

Example 2

bessely(3:9,(0:.2:10)') generates the entire table on page 399 of [1] Abramowitz and Stegun, Handbook of Mathematical Functions.

Algorithms

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

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.

See Also

besselh | besseli | besselj | besselk

© 1984-2012- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS