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# bessely

Bessel function of the second kind

bessely(nu,z)
bessely(nu,A)

## Description

bessely(nu,z) returns the Bessel function of the second kind, Yν(z).

bessely(nu,A) returns the Bessel function of the second kind for each element of A.

## Input Arguments

 nu Symbolic number, variable, or expression representing a real number. z Symbolic number, variable, or expression. A Vector or matrix of symbolic numbers, variables, or expressions.

## Examples

Solve this second-order differential equation. The solutions are the Bessel functions of the first and the second kind.

```syms nu w(z)
dsolve(z^2*diff(w, 2) + z*diff(w) +(z^2 - nu^2)*w == 0)```
```ans =
C2*besselj(nu, z) + C3*bessely(nu, z)```

Verify that the Bessel function of the second kind is a valid solution of the Bessel differential equation:

```syms nu z
simplify(z^2*diff(bessely(nu, z), z, 2) + z*diff(bessely(nu, z), z) + (z^2 - nu^2)*bessely(nu, z)) == 0```
```ans =
1```

Compute the Bessel functions of the second kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

`[bessely(0, 5), bessely(-1, 2), bessely(1/3, 7/4),  bessely(1, 3/2 + 2*i)]`
```ans =
-0.3085             0.1070             0.2358            -0.4706 + 1.5873i```

Compute the Bessel functions of the second kind for the numbers converted to symbolic objects. For most symbolic (exact) numbers, bessely returns unresolved symbolic calls.

`[bessely(sym(0), 5), bessely(sym(-1), 2), bessely(1/3, sym(7/4)),  bessely(sym(1), 3/2 + 2*i)]`
```ans =
[ bessely(0, 5), -bessely(1, 2), bessely(1/3, 7/4), bessely(1, 3/2 + 2*i)]```

For symbolic variables and expressions, bessely also returns unresolved symbolic calls:

```syms x y
[bessely(x, y), bessely(1, x^2), bessely(2, x - y), bessely(x^2, x*y)]```
```ans =
[ bessely(x, y), bessely(1, x^2), bessely(2, x - y), bessely(x^2, x*y)]```

If the first parameter is an odd integer multiplied by 1/2, besseli rewrites the Bessel functions in terms of elementary functions:

```syms x
bessely(1/2, x)```
```ans =
-(2^(1/2)*cos(x))/(pi^(1/2)*x^(1/2))```
`bessely(-1/2, x)`
```ans =
(2^(1/2)*sin(x))/(pi^(1/2)*x^(1/2))```
`bessely(-3/2, x)`
```ans =
(2^(1/2)*(cos(x) - sin(x)/x))/(pi^(1/2)*x^(1/2))```
`bessely(5/2, x)`
```ans =
-(2^(1/2)*((3*sin(x))/x + cos(x)*(3/x^2 - 1)))/(pi^(1/2)*x^(1/2))```

Differentiate the expressions involving the Bessel functions of the second kind:

```syms x y
diff(bessely(1, x))
diff(diff(bessely(0, x^2 + x*y -y^2), x), y)```
```ans =
bessely(0, x) - bessely(1, x)/x

ans =
- bessely(1, x^2 + x*y - y^2) -...
(2*x + y)*(bessely(0, x^2 + x*y - y^2)*(x - 2*y) -...
(bessely(1, x^2 + x*y - y^2)*(x - 2*y))/(x^2 + x*y - y^2))```

Call bessely for the matrix A and the value 1/2. The result is a matrix of the Bessel functions bessely(1/2, A(i,j)).

```syms x
A = [-1, pi; x, 0];
bessely(1/2, A)```
```ans =
[          (2^(1/2)*cos(1)*i)/pi^(1/2), 2^(1/2)/pi]
[ -(2^(1/2)*cos(x))/(pi^(1/2)*x^(1/2)),        Inf]```

Plot the Bessel functions of the second kind for ν = 0, 1, 2, 3:

```syms x y
for nu =[0, 1, 2, 3]
ezplot(bessely(nu, x) - y, [0, 10, -1, 0.6])
colormap([0 0 1])
hold on
end
title('Bessel functions of the second kind')
ylabel('besselY(x)')
grid
hold off```

expand all

### Bessel Functions of the Second Kind

The Bessel differential equation

has two linearly independent solutions. These solutions are represented by the Bessel functions of the first kind, Jν(z), and the Bessel functions of the second kind, Yν(z):

The Bessel functions of the second kind are defined via the Bessel functions of the first kind:

Here Jν(z) are the Bessel function of the first kind:

### Tips

• Calling bessely for a number that is not a symbolic object invokes the MATLAB® bessely function.

## References

[1] Olver, F. W. J. "Bessel Functions of Integer Order." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

[2] Antosiewicz, H. A. "Bessel Functions of Fractional Order." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.