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besselk - Modified Bessel function of second kind

Syntax

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

Definitions

The differential equation

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

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

where I_ν(z) and I_–ν(z) form a fundamental set of solutions of the modified Bessel's equation for noninteger ν

and Γ(a) is the gamma function. K_ν(z) is independent of I_ν(z).

I_ν(z) can be computed using besseli.

Description

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

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` successfully 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].

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.