besseli

Modified Bessel function of the first kind for symbolic expressions

Syntax

besseli(nu,z)

Description

besseli(nu,z) returns the modified Bessel function of the first kind, I_ν(z).

Examples

Find Modified Bessel Function of First Kind

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)]

Solve Bessel Differential Equation for Modified Bessel Functions

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 =
  logical
   1

Special Values of Modified Bessel Function of First Kind

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))/(x^(1/2)*pi^(1/2))

besseli(-1/2, x)

ans =
(2^(1/2)*cosh(x))/(x^(1/2)*pi^(1/2))

besseli(-3/2, x)

ans =
(2^(1/2)*(sinh(x) - cosh(x)/x))/(x^(1/2)*pi^(1/2))

besseli(5/2, x)

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

Differentiate Modified Bessel Function of First Kind

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))
 

Bessel Function for Matrix Input

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

Plot the Modified Bessel Functions of the First Kind

Plot the modified Bessel functions of the first kind for $$v=0,1,2,3$$.

syms x y
fplot(besseli(0:3, x))
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')

Input Arguments

nu — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, array, or a symbolic number, variable, expression, function, or array. If nu is a vector or matrix, besseli returns the modified Bessel function of the first kind for each element of nu.

z — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, array, or a symbolic number, variable, expression, function, or array. If nu is a vector or matrix, besseli returns the modified Bessel function of the first kind for each element of nu.

More About

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, I_ν(z), and the modified Bessel functions of the second kind, K_ν(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:

$$I_{\nu }\left(z\right)=\frac{{\left(z/2\right)}^{\nu }}{\sqrt{\pi }\Gamma \left(\nu +1/2\right)}{\displaystyle \underset{0}{\overset{\pi }{\int }}{e}^{z\cos \left(t\right)}\sin {\left(t\right)}^{2\nu }dt}$$

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.

Version History

Introduced in R2014a