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# bessely

Bessel function of second kind

## Syntax

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

## Definitions

The differential equation

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

A solution Yν(z) of the second kind can be expressed as

where Jν(z) and Jν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν

and Γ(a) is the gamma function. Yν(z) is linearly independent of Jν(z).

Jν(z) can be computed using besselj.

## Description

Y = bessely(nu,Z) computes Bessel functions of the second kind, Yν(z), 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.

Y = bessely(nu,Z,1) computes bessely(nu,Z).*exp(-abs(imag(Z))).

## Examples

expand all

### Vector of Function Values

Create a column vector of domain values.

```z = (0:0.2:1)';
```

Calculate the function values using bessely with nu = 1.

```format long
bessely(1,z)
```
```ans =

-Inf
-3.323824988111848
-1.780872044270052
-1.260391347177388
-0.978144176683359
-0.781212821300289```

### Plot Bessel Functions of Second Kind

Define the domain.

```X = 0:0.1:20;
```

Calculate the first five Bessel functions of the second kind.

```Y = zeros(5,201);
for i=0:4
Y(i+1,:) = bessely(i,X);
end
```

Plot the results.

```plot(X,Y,'LineWidth',1.5)
axis([-0.1 20.2 -2 0.6])
grid on;
legend('Y_0','Y_1','Y_2','Y_3','Y_4','Location','Best')
title('Bessel Functions of the Second Kind for v = 0,1,2,3,4')
xlabel('X')
ylabel('Y_v(X)')
```

expand all

### Tips

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

where is besselh, Jν(z) is besselj, and Yν(z) is bessely. The Hankel functions also form a fundamental set of solutions to Bessel's equation (see besselh).

### Algorithms

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

## 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.