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)

 

image\bessj.gif

 

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:

 

image\bessy01.gif

 

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:

 

image\bessj01.gif

 

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.

See Also:

AIRY

BESSELH

BESSELI

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