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Modified Bessel function of the second kind



The differential equation

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

A solution of the second kind can be expressed as

where and form a fundamental set of solutions of the modified Bessel's equation for noninteger

and is the gamma function. is independent of .

can be computed using besseli.


K = besselk(nu,Z) computes the modified Bessel function 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.

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.

besselk successfully computed the modified Bessel function for this element.
Illegal arguments.
Overflow. Returns Inf.
Some loss of accuracy in argument reduction.
Unacceptable loss of accuracy, Z or nu too large.
No convergence. Returns NaN.


Example 1.

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.


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

See Also

airy, besselh, besseli, besselj, bessely


[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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