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# besselj

Bessel function of the first kind

## Syntax

```besselj(nu,z) ```

## Description

`besselj(nu,z)` returns the Bessel function of the first kind, Jν(z).

## Input Arguments

 `nu` Symbolic number, variable, expression, function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If `nu` is a vector or matrix, `besseli` returns the Bessel function of the first kind for each element of `nu`. `z` Symbolic number, variable, expression, or function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If `z` is a vector or matrix, `besseli` returns the Bessel function of the first kind for each element of `z`.

## Examples

### Find Bessel Function of First Kind

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

```[besselj(0, 5), besselj(-1, 2), besselj(1/3, 7/4),... besselj(1, 3/2 + 2*i)]```
```ans = -0.1776 + 0.0000i -0.5767 + 0.0000i 0.5496 + 0.0000i 1.6113 + 0.3982i```

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

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

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

```syms x y [besselj(x, y), besselj(1, x^2), besselj(2, x - y), besselj(x^2, x*y)]```
```ans = [ besselj(x, y), besselj(1, x^2), besselj(2, x - y), besselj(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 first kind is a valid solution of the Bessel differential equation:

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

### Special Values of Bessel Function of First Kind

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

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

### Differentiate Bessel Function of First Kind

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

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

### Find Bessel Function for Matrix Input

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

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

### Plot Bessel Functions of First Kind

Plot the Bessel functions of the first kind for . Prior to R2016a, use `ezplot` instead of `fplot`.

```syms x y fplot(besselj(0:3, x)) axis([0 10 -0.5 1.1]) grid on ylabel('J_v(x)') legend('J_0','J_1','J_2','J_3', 'Location','Best') title('Bessel functions of the first kind')```

collapse all

### Bessel Functions of the First 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):

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

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

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