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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

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
simplify(z^2*diff(besselj(nu, z), z, 2) + z*diff(besselj(nu, z), z) + (z^2 - nu^2)*besselj(nu, z)) == 0
ans =
     1
 

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            -0.5767             0.5496             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 + 2*i)]

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)]
 

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

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))
 

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)*i)/pi^(1/2), 0]
[ (2^(1/2)*sin(x))/(pi^(1/2)*x^(1/2)), 0]
 

Plot the Bessel functions of the first kind for ν = 0, 1, 2, 3:

syms x y
for nu =[0, 1, 2, 3]
  ezplot(besselj(nu, x) - y, [0, 10, -0.5, 1.1])
  colormap([0 0 1])
  hold on
end
title('Bessel functions of the first kind')
ylabel('besselJ(x)')
grid
hold off

More About

expand all

Bessel Functions of the First Kind

The Bessel differential equation

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):

This formula is the integral representation of the Bessel functions of the first kind:

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.

See Also

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