besseli

Modified Bessel function of the first kind

besseli(nu,z)

Description

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

Examples

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 =
1

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)]

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

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))

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

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

syms x y
for nu = [0, 1, 2, 3]
ezplot(besseli(nu, x))
hold on
end
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')
hold off

collapse all

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.