BESSELY

Purpose:

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

Syntax:

BESSELY(v, z, opt)

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

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 Y_ν(z) where ν is the order and z is the input.

Example:

bessely(0, 3)

returns 0.376850, the value of Y₀(3).

Example:

W1: 0..0.2..1;

W2: bessely(1, w1)

Returns Y₁(z) for z between 0 and 1. W2 contains the series

{-inf, -3.323825, -1.780872, -1.260391, -0.978144, -0.781213}

Example:

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

Evaluates the Bessel function of the second 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, Y_ν(z) is a solution of the second kind and J_ν(z) is a linearly independent solution of the first kind For non-integer order ν:

$image\bessy02.gif$

See YN for a faster evaluation of the Bessel function of the second kind for strictly integer orders.

If z is real and z < 0, the result is generally complex

See BESSELJ to evaluate J_ν(z), the Bessel function of the first kind.

BESSELY 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

BESSELY

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