MATLAB Function Reference |

Bessel functions of the second kind

**Syntax**

**Definition**

where is a real constant, is called *Bessel's equation*, and its solutions are known as *Bessel functions*.

A solution of the second kind can be expressed as

where and form a fundamental set of solutions of Bessel's equation for noninteger

and is the gamma function. is linearly independent of

can be computed using `besselj`

.

**Description**

```
Y = bessely(nu,Z)
```

computes Bessel functions of the second kind, , for each element of the array `Z`

. The order `nu`

need not be an integer, but must be real. The argument `Z`

can be complex. The result is real where `Z`

is positive.

If `nu`

and `Z`

are arrays of the same size, the result is also that size. If either input is a scalar, it is expanded to the other input's size. If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.

```
Y = bessely(nu,Z,1)
```

computes `bessely(nu,Z).*exp(-abs(imag(Z)))`

.

```
[Y,ierr] = bessely(nu,Z)
```

also returns completion flags in an array the same size as `Y`

.

**Remarks**

The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,

where is `besselh`

, is `besselj`

, and is `bessely`

. The Hankel functions also form a fundamental set of solutions to Bessel's equation (see `besselh`

).

**Examples**

format long z = (0:0.2:1)'; bessely(1,z) ans = -Inf -3.32382498811185 -1.78087204427005 -1.26039134717739 -0.97814417668336 -0.78121282130029

**Example 2.** `bessely(3:9,(0:.2:10)')`

generates the entire table on page 399 of [1] Abramowitz and Stegun, *Handbook of Mathematical Functions.*

**Algorithm**

The `bessely`

function uses a Fortran MEX-file to call a library developed by D. E Amos [3] [4].

**See Also**

`besselh`

, `besseli`

, `besselj`

, `besselk`

**References**

[1] Abramowitz, M. and I.A. Stegun, *Handbook of Mathematical Functions*,
National Bureau of Standards, Applied Math. Series #55, Dover Publications,
1965, sections 9.1.1, 9.1.89 and 9.12, formulas 9.1.10 and 9.2.5.

[2] Carrier, Krook, and Pearson, *Functions of a Complex Variable: Theory and
Technique*, Hod Books, 1983, section 5.5.

[3] Amos, D. E., "A Subroutine Package for Bessel Functions of a Complex
Argument and Nonnegative Order," *Sandia National Laboratory Report*,
SAND85-1018, May, 1985.

[4] Amos, D. E., "A Portable Package for Bessel Functions of a Complex
Argument and Nonnegative Order," *Trans. Math. Software*, 1986.

besselk | beta |

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