# bessely

Bessel function of the second kind for symbolic expressions

## Syntax

``bessely(nu,z)``

## Description

example

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

## Examples

### Find Bessel Function of Second Kind

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 + 0.0000i 0.1070 + 0.0000i 0.2358 + 0.0000i -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 + 2i)]```

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)]```

### Solve Bessel Differential Equation for Bessel Functions

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 isAlways(z^2*diff(bessely(nu, z), z, 2) + z*diff(bessely(nu, z), z)... + (z^2 - nu^2)*bessely(nu, z) == 0)```
```ans = logical 1```

### Special Values of Bessel Function of Second Kind

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

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

### Differentiate Bessel Functions of Second Kind

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))```

### Find Bessel Function for Matrix Input

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)*1i)/pi^(1/2), 2^(1/2)/pi] [ -(2^(1/2)*cos(x))/(x^(1/2)*pi^(1/2)), Inf]```

### Plot Bessel Functions of Second Kind

Plot the Bessel functions of the second kind for $v=0,1,2,3$.

```syms x y fplot(bessely(0:3,x)) axis([0 10 -1 0.6]) grid on ylabel('Y_v(x)') legend('Y_0','Y_1','Y_2','Y_3', 'Location','Best') title('Bessel functions of the second kind')```

## Input Arguments

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Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

If `nu` is a vector or matrix, `bessely` returns the Bessel function of the second kind for each element of `nu`.

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

If `z` is a vector or matrix, `bessely` returns the Bessel function of the second kind for each element of `z`.

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### Bessel Function of the Second Kind

The Bessel differential equation

`${z}^{2}\frac{{d}^{2}w}{d{z}^{2}}+z\frac{dw}{dz}+\left({z}^{2}-{\nu }^{2}\right)w=0$`

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):

`$w\left(z\right)={C}_{1}{J}_{\nu }\left(z\right)+{C}_{2}{Y}_{\nu }\left(z\right)$`

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

`${Y}_{\nu }\left(z\right)=\frac{{J}_{\nu }\left(z\right)\mathrm{cos}\left(\nu \pi \right)-{J}_{-\nu }\left(z\right)}{\mathrm{sin}\left(\nu \pi \right)}$`

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

`${J}_{\nu }\left(z\right)=\frac{{\left(z/2\right)}^{\nu }}{\sqrt{\pi }\Gamma \left(\nu +1/2\right)}\underset{0}{\overset{\pi }{\int }}\mathrm{cos}\left(z\mathrm{cos}\left(t\right)\right)\mathrm{sin}{\left(t\right)}^{2\nu }dt$`

## Tips

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

At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, `bessely(nu,z)` expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

## 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.