besselj
Bessel function of the first kind for symbolic expressions
Syntax
Description
besselj(
returns the
Bessel function of the first kind, Jν(z).nu
,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 .
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
More About
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