# BESSELI

## Purpose:

Evaluates the modified Bessel function of the first kind for any real order.

## Syntax:

BESSELI(v, z, opt)

v

-

A real or real series, the order. The order must be real but need not be an integer.

z

-

Any scalar or series. The input value.

opt

-

Optional. An integer, the scaling method:

 0: no scaling (default) 1: scale by e-mag(real(z))

## Returns:

A scalar or series, the value of Iν(z) where ν is the order and z is the input.

## Example:

besseli(0, 3)

returns 0.260052, the value of I0(3).

## Example:

W1: 0..0.2..1;

W2: besseli(1, w1)

Returns I1(z) for z between 0 and 1. W2 contains the series

{0.0, 0.100501, 0.204027, 0.313704, 0.432865, 0.565159}

## Example:

besseli((3..9)', 0..0.2..10)

Evaluates the modified Bessel function of the first kind for orders 3 through 9 with inputs from 0 to 10. Each column of the result contains the output for the specified order.

## Remarks:

Modified Bessel functions are solutions to the differential equation:

where ν is the order, Iν(z) is a solution of the first kind and Kν(z) is a linearly independent solution of the second kind. For non-integer order ν:

If the order is not an integer, and z is real where z < 0, the result is generally complex

See BESSELK to evaluate Kν(z), the modified Bessel function of the second kind.

BESSELI is based on a FORTRAN library written by D. E. Amos.

AIRY

BESSELH

BESSELJ

BESSELK

BESSELY

JN

YN

## References:

[1]   Abramowitz and Stegun

Handbook of Mathematical Functions (9th printing 1970)

US Gov. Printing Office

Section 9.1.1, 9.1.89, 9.12

[2]   Amos, D.E.

A Subroutine Package for Bessel Functions of a Complex

Argument and Nonnegative Order

Sandia National Laboratory Report

SAND85-1018, May, 1985.

[3]   Amos, D.E.

A Portable Package for Bessel Functions of a Complex

Argument and Nonnegative Order

Trans. Math. Software, 1986