MATLAB Function Reference

besseli, besselk

Modified Bessel functions

Syntax

I = besseli(nu,Z)     Modified Bessel function of the 1st kind
K = besselk(nu,Z)     Modified Bessel function of the 2nd kind
I = besseli(nu,Z,1)
K = besselk(nu,Z,1)
[I,ierr] = besseli(...)
[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.

and form a fundamental set of solutions of the modified Bessel's equation for noninteger . is a second solution, independent of .

and are defined by:

Description

I = besseli(nu,Z) computes modified Bessel functions 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.

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

I = besseli(nu,Z,1) computes besseli(nu,Z).*exp(-abs(real(Z))).

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

[I,ierr] = besseli(...) and [K,ierr] = besselk(...) 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.

Examples

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

besseli(1,z)

ans =
                  0
   0.10050083402813
   0.20402675573357
   0.31370402560492
   0.43286480262064
   0.56515910399249

besselk(1,z)

ans =
                Inf
   4.77597254322047
   2.18435442473269
   1.30283493976350
   0.86178163447218
   0.60190723019723

besseli(3:9,(0:.2,10)',1) generates the entire table on page 423 of Abramowitz and Stegun, Handbook of Mathematical Functions.

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

Algorithm

The besseli and besselk functions use a Fortran MEX-file to call a library developed by D. E. Amos [3] [4].

See Also

airy, besselj, bessely

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.

besselh

besselj, bessely