MATLAB Function Reference |
Bessel functions of the second kind
Syntax
Definition
where is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions.
A solution of the second kind can be expressed as
where and form a fundamental set of solutions of Bessel's equation for noninteger
and is the gamma function. is linearly independent of
can be computed using besselj
.
Description
Y = bessely(nu,Z)
computes Bessel functions 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.
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
.
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
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.
Algorithm
The bessely
function uses a Fortran MEX-file to call a library developed by D. E Amos [3] [4].
See Also
besselh
, besseli
, besselj
, 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.
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