besselk

Modified Bessel function of the second kind for symbolic expressions

Syntax

besselk(nu,z)

Description

besselk(nu,z) returns the modified Bessel function of the second kind, K_ν(z).

Examples

Find Modified Bessel Function of Second Kind

Compute the modified Bessel functions of the second kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

[besselk(0, 5), besselk(-1, 2), besselk(1/3, 7/4),...
  besselk(1, 3/2 + 2*i)]

ans =
   0.0037 + 0.0000i   0.1399 + 0.0000i   0.1594 + 0.0000i  -0.1620 - 0.1066i

Compute the modified Bessel functions of the second kind for the numbers converted to symbolic objects. For most symbolic (exact) numbers, besselk returns unresolved symbolic calls.

[besselk(sym(0), 5), besselk(sym(-1), 2),...
 besselk(1/3, sym(7/4)), besselk(sym(1), 3/2 + 2*i)]

ans =
[ besselk(0, 5), besselk(1, 2), besselk(1/3, 7/4), besselk(1, 3/2 + 2i)]

For symbolic variables and expressions, besselk also returns unresolved symbolic calls:

syms x y
[besselk(x, y), besselk(1, x^2), besselk(2, x - y), besselk(x^2, x*y)]

ans =
[ besselk(x, y), besselk(1, x^2), besselk(2, x - y), besselk(x^2, x*y)]

Special Values of Modified Bessel Function of Second Kind

If the first parameter is an odd integer multiplied by 1/2, besselk rewrites the Bessel functions in terms of elementary functions:

syms x
besselk(1/2, x)

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

besselk(-1/2, x)

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

besselk(-3/2, x)

ans =
(2^(1/2)*pi^(1/2)*exp(-x)*(1/x + 1))/(2*x^(1/2))

besselk(5/2, x)

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

Solve Bessel Differential Equation for 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 second kind is a valid solution of the modified Bessel differential equation:

syms nu z
isAlways(z^2*diff(besselk(nu, z), z, 2) + z*diff(besselk(nu, z), z)...
 - (z^2 + nu^2)*besselk(nu, z) == 0)

ans =
  logical
   1

Differentiate Modified Bessel Function of Second Kind

Differentiate the expressions involving the modified Bessel functions of the second kind:

syms x y
diff(besselk(1, x))
diff(diff(besselk(0, x^2 + x*y -y^2), x), y)

ans =
- besselk(1, x)/x - besselk(0, x)
 
ans =
(2*x + y)*(besselk(0, x^2 + x*y - y^2)*(x - 2*y) +...
(besselk(1, x^2 + x*y - y^2)*(x - 2*y))/(x^2 + x*y - y^2)) -...
besselk(1, x^2 + x*y - y^2)
 

Find Bessel Function for Matrix Input

Call besselk for the matrix A and the value 1/2. The result is a matrix of the modified Bessel functions besselk(1/2, A(i,j)).

syms x
A = [-1, pi; x, 0];
besselk(1/2, A)

ans =
[         -(2^(1/2)*pi^(1/2)*exp(1)*1i)/2, (2^(1/2)*exp(-pi))/2]
[ (2^(1/2)*pi^(1/2)*exp(-x))/(2*x^(1/2)),                  Inf]

Plot Modified Bessel Functions of Second Kind

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

syms x y
fplot(besselk(0:3, x))
axis([0 4 0 4])
grid on

ylabel('K_v(x)')
legend('K_0','K_1','K_2','K_3', 'Location','Best')
title('Modified Bessel functions of the second 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 Second 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):

$$w\left(z\right)=C_1{I}_{\nu }\left(z\right)+C_2{K}_{\nu }\left(z\right)$$

The modified Bessel functions of the second kind are defined via the modified Bessel functions of the first kind:

$$K_{\nu }\left(z\right)=\frac{\pi /2}{\sin \left(\nu \pi \right)}\left(I_{-\nu }\left(z\right)-I_{\nu }\left(z\right)\right)$$

Here I_ν(z) are 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 besselk for a number that is not a symbolic object invokes the MATLAB^® besselk 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, besselk(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