IDL Analyst Reference Guide: Special Functions

IMSL_BESSI

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

Syntax

Result = IMSL_BESSI(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_BESSI, where n = N_ELEMENTS(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_BESSI function evaluates a modified Bessel function of the first kind with real order and real or complex parameters. The data type of the returned value is always complex.

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

For large parameters, z, Temme's (1975) algorithm is used to find I_v(z). The I_v(z) values are recurred upward (if stable). This involves evaluating a continued fraction. If this evaluation fails to converge, the answer may not be accurate. For moderate and small parameters, Miller's method is used.

Example

In this example, J_{0.3 + v–1}(1.2 + 0.5i), v = 1, ... 4 is computed and printed first by calling IMSL_BESSI four times in a row, then by using the keyword SEQUENCE.

z = COMPLEX(1.2, .5)  
FOR i = 0, 3 DO PM, IMSL_BESSI(i + .3, z)  
   (    1.16339,  0.396301)  
   (   0.447264,  0.332142)  
   (  0.0821799,  0.127165)  
   ( 0.00577678, 0.0286277)  
PM, IMSL_BESSI(.3, z, Sequence = 4), Title = 'With SEQUENCE:'  
With SEQUENCE:  
   (    1.16339,  0.396301)  
   (   0.447264,  0.332142)  
   (  0.0821799,  0.127165)  
   ( 0.00577678, 0.0286277)