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### IMSL_BESSK

IMSL_BESSK

The IMSL_BESSK function evaluates a modified Bessel function of the second kind with real order and real or complex parameters.

## Syntax

Result = IMSL_BESSK(order, z [, /DOUBLE] [, SEQUENCE=value])

## Return Value

The desired value of the modified Bessel function.

## Arguments

#### order

Real parameter specifying the desired order. The argument order must be greater than –1/2.

#### z

Real or complex parameter for which the Bessel function is to be evaluated.

## Keywords

#### DOUBLE

If present and nonzero, double precision is used.

#### SEQUENCE

If present and nonzero, a one-dimensional array of length n containing the values of the Bessel function through the series is returned by IMSL_BESSK, where n = NELEMENTS(SEQUENCE). The i-th element of this array is the Bessel function of order (order + i) at z for i = 0, ... (n – 1).

## Discussion

The IMSL_BESSK function evaluates a modified Bessel function of the second kind with real order and real or complex parameters. The data type of the returned value is always complex.

The Bessel function, Kv(z), is defined as follows:

This function is based on the code BESSCC of Thompson (1981) and Thompson and Barnett (1987). For moderate or large parameters, z, Temme's (1975) algorithm is used to find Kv (z). This involves evaluating a continued fraction. If this evaluation fails to converge, the answer may not be accurate. For small z, a Neumann series is used to compute Kv (z). Upward recurrence of the Kv (z) is always stable.

## Example

In this example, K0.3 + v–1(1.2 + 0.5i), v = 1, ..., 4 is computed and printed.

`z = COMPLEX(1.2, .5)  FOR i = 0, 3 DO PM, IMSL_BESSK(i + .3, z)     ( 0.245546, -0.199599)     ( 0.335637, -0.362005)     ( 0.586718,  -1.12610)     ( 0.719457,  -4.83864)  PM, IMSL_BESSK(.3, z, Sequence = 4), Title = ';With SEQUENCE:'  With SEQUENCE:     ( 0.245546, -0.199599)     ( 0.335637, -0.362005)     ( 0.586718,  -1.12610)     ( 0.719456,  -4.83864)  `

## Version History

 6.4 Introduced