# besselk

Modified Bessel function of the second kind for symbolic expressions

## Syntax

``besselk(nu,z)``

## Description

example

````besselk(nu,z)` returns the modified Bessel function of the second kind, Kν(z).```

## Examples

### Find Modified Bessel Function of Second Kind

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

```[besselk(0, 5), besselk(-1, 2), besselk(1/3, 7/4),... besselk(1, 3/2 + 2*i)]```
```ans = 0.0037 + 0.0000i 0.1399 + 0.0000i 0.1594 + 0.0000i -0.1620 - 0.1066i```

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

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

For symbolic variables and expressions, `besselk` also returns unresolved symbolic calls:

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

### Special Values of Modified Bessel Function of Second Kind

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

```syms x besselk(1/2, x)```
```ans = (2^(1/2)*pi^(1/2)*exp(-x))/(2*x^(1/2))```
`besselk(-1/2, x)`
```ans = (2^(1/2)*pi^(1/2)*exp(-x))/(2*x^(1/2))```
`besselk(-3/2, x)`
```ans = (2^(1/2)*pi^(1/2)*exp(-x)*(1/x + 1))/(2*x^(1/2))```
`besselk(5/2, x)`
```ans = (2^(1/2)*pi^(1/2)*exp(-x)*(3/x + 3/x^2 + 1))/(2*x^(1/2))```

### Solve Bessel Differential Equation for Bessel Functions

Solve this second-order differential equation. The solutions are the modified 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*besseli(nu, z) + C3*besselk(nu, z)```

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

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

### Differentiate Modified Bessel Function of Second Kind

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

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

### Find Bessel Function for Matrix Input

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

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

### Plot Modified Bessel Functions of Second Kind

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

```syms x y fplot(besselk(0:3, x)) axis([0 4 0 4]) grid on ylabel('K_v(x)') legend('K_0','K_1','K_2','K_3', 'Location','Best') title('Modified Bessel functions of the second kind')```

## Input Arguments

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Input, specified as a number, vector, matrix, array, or a symbolic number, variable, expression, function, or array. If `nu` is a vector or matrix, `besseli` returns the modified Bessel function of the first kind for each element of `nu`.

Input, specified as a number, vector, matrix, array, or a symbolic number, variable, expression, function, or array. If `nu` is a vector or matrix, `besseli` returns the modified Bessel function of the first kind for each element of `nu`.

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### Modified Bessel Functions of the Second Kind

The modified 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 modified Bessel functions of the first kind, Iν(z), and the modified Bessel functions of the second kind, Kν(z):

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

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

`${K}_{\nu }\left(z\right)=\frac{\pi /2}{\mathrm{sin}\left(\nu \pi \right)}\left({I}_{-\nu }\left(z\right)-{I}_{\nu }\left(z\right)\right)$`

Here Iν(z) are the modified Bessel functions of the first kind:

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

## Tips

• Calling `besselk` for a number that is not a symbolic object invokes the MATLAB® `besselk` 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, `besselk(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.