# BESSELY

## Purpose:

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

## Syntax:

BESSELY(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 Yν(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:

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

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 Jν(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