# besseli

Modified Bessel function of the first kind for symbolic expressions

## Syntax

``besseli(nu,z)``

## Description

example

````besseli(nu,z)` returns the modified Bessel function of the first kind, Iν(z).```

## Examples

### Find Modified Bessel Function of First Kind

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

`[besseli(0, 5), besseli(-1, 2), besseli(1/3, 7/4), besseli(1, 3/2 + 2*i)]`
```ans = 27.2399 + 0.0000i 1.5906 + 0.0000i 1.7951 + 0.0000i -0.1523 + 1.0992i```

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

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

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

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

### Solve Bessel Differential Equation for Modified 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 first kind is a valid solution of the modified Bessel differential equation.

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

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

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

```syms x besseli(1/2, x)```
```ans = (2^(1/2)*sinh(x))/(x^(1/2)*pi^(1/2))```
`besseli(-1/2, x)`
```ans = (2^(1/2)*cosh(x))/(x^(1/2)*pi^(1/2))```
`besseli(-3/2, x)`
```ans = (2^(1/2)*(sinh(x) - cosh(x)/x))/(x^(1/2)*pi^(1/2))```
`besseli(5/2, x)`
```ans = -(2^(1/2)*((3*cosh(x))/x - sinh(x)*(3/x^2 + 1)))/(x^(1/2)*pi^(1/2))```

### Differentiate Modified Bessel Function of First Kind

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

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

### Bessel Function for Matrix Input

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

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

### Plot the Modified Bessel Functions of the First Kind

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

```syms x y fplot(besseli(0:3, x)) axis([0 4 -0.1 4]) grid on ylabel('I_v(x)') legend('I_0','I_1','I_2','I_3', 'Location','Best') title('Modified Bessel functions of the first 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 First 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):

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

This formula is the integral representation of 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 `besseli` for a number that is not a symbolic object invokes the MATLAB® `besseli` 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, `besseli(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.

## Version History

Introduced in R2014a