# besselj

Bessel function of the first kind for symbolic expressions

## Description

example

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

## Examples

### Find Bessel Function of First Kind

Compute the Bessel functions of the first kind for these numbers. Because these numbers are floating point, 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 form. 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)
ode = z^2*diff(w,2) + z*diff(w) +(z^2-nu^2)*w == 0;
dsolve(ode)
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.

cond = subs(ode,w,besselj(nu,z));
isAlways(cond)
ans =
logical
1

### Special Values of Bessel Function of First Kind

Show that 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 expressions involving the Bessel functions of the first kind.

syms x y
diff(besselj(1,x))
ans =
besselj(0, x) - besselj(1, x)/x
diff(diff(besselj(0,x^2+x*y-y^2), x), y)
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. besselj acts element-wise to return matrix of Bessel functions.

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 $0,1,2,3$.

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')

## 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, besselj returns the modified Bessel function of the first 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 nu is a vector or matrix, besselj returns the modified Bessel function of the first kind for each element of nu.

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### Bessel Functions of the First Kind

The Bessel functions are solutions of the Bessel differential equation.

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

These solutions are 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.

## Version History

Introduced in R2014a