MATLAB Function Reference |

Modified Bessel function of the second kind

**Syntax**

**Definitions**

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`

.

**Description**

```
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`

.

**Examples**

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].

**See Also**

`airy`

, `besselh`

, `besseli`

, `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.

besselj | bessely |