# BESSELJ

## Purpose:

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

## Syntax:

BESSELJ(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(imag(z))

## Returns:

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

## Example:

besselj(0, 3)

returns 0.260052, the value of J0(3).

## Example:

W1: 0..0.2..1;

W2: besselj(1, w1)

Returns J1(z) for z between 0 and 1. W2 contains the series

{0.0, 0.099501, 0.196027, 0.286701, 0.368842, 0.440051}

## Example:

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

Evaluates the 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:

Bessel functions are solutions to the differential equation:

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

See JN for a faster evaluation of the Bessel function of the first kind for strictly integer orders.

If the order is not an integer, the result is generally complex.

See BESSELY to evaluate Yν(z), the Bessel function of the second kind.

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

## 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