# Documentation/Calc Functions/BESSELK

BESSELK

## Summary:

Calculates the value of a modified Bessel function of the second kind, usually referred to as Kn(x). n is the order of the function and x is the value at which the function is to be calculated.

BESSELK(X; N)

## Returns:

Returns a real number which is the value of the modified Bessel function of the second kind, with a specified order, for a given value of x.

## Arguments:

X is a positive real number, or a reference to a cell containing that number, which is the value at which the modified Bessel function of the second kind is to be calculated.

N is a non-negative integer, or a reference to a cell containing that number, which is the order of the modified Bessel function of the second kind.

• If either X or N is non-numeric, then BESSELK reports a #VALUE! error.
• If X is not greater than 0, then BESSELK reports an invalid argument error (Err:502).
• If N is a non-integer value, then BESSELK truncates it to an integer.
• If N is negative, then BESSELK reports an invalid argument error (Err:502).

The following figure shows a plot of the modified Bessel functions of the second kind for orders 0, 1, 2, and 3.

Modified Bessel functions of the second kind are singular at the origin, x = 0.

ODF 1.2 indicates that the first argument to the BESSELK function should be an integer. However, this is considered an unnecessary limitation and Calc’s implementation caters for positive real values of X.

### General information about Calc's Bessel functions

 For convenience, the information in this subsection is repeated on all pages describing Calc’s Bessel and modified Bessel functions.

Calc provides two Bessel functions that are solutions of Bessel's differential equation and two modified Bessel functions that are solutions of the modified Bessel equation. The mathematics behind these equations and functions is beyond the scope of this wiki but more information can be found at Wikipedia’s Bessel function page and in the sources that it references.

The four functions provided by Calc are as follows:

• BESSELJ – calculates values of Jn(x), a Bessel function of the first kind, for non-negative integer vales of n.
• BESSELY – calculates values of Yn(x), a Bessel function of the second kind, for non-negative integer vales of n and positive values of x.
• BESSELI – calculates values of In(x), a modified Bessel function of the first kind, for non-negative integer vales of n.
• BESSELK - calculates values of Kn(x), a modified Bessel function of the second kind, for non-negative integer vales of n and positive values of x.

In each case the parameter n is known as the order of the function.

## Examples:

Formula Description Returns
=BESSELK(3.45; 4) Here the function calculates the value of the modified Bessel function of the second kind, with order 4, at an x value of 3.45. 0.144803466373734
=BESSELK(3.45; 4.333) Here the function calculates the value of the modified Bessel function of the second kind, with order 4.333, at an x value of 3.45. The result is the same as in the previous example, showing that Calc truncates the value of 4.333 to 4. 0.144803466373734
=BESSELK(0; 4) Here the function returns an error, because the x value must be greater than 0. Err:502
=BESSELK(H1; H2) where cells H1 and H2 contain the values 2.6 and 3 respectively. Here the function calculates the value of the modified Bessel function of the second kind, with order 3, at an x value of 2.6. 0.227771428410702

BESSELK