besselk

Modified Bessel function of the second kind

Syntax

besselk(nu,z)

Description

besselk(nu,z) returns the modified Bessel function of the second kind, Kν(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 second 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 second 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 second kind is a valid solution of the modified Bessel differential equation:

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

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 + 2*i)]

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)]

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))

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)
 

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

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

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

More About

expand all

Modified Bessel Functions of the Second Kind

The modified Bessel differential equation

z2d2wdz2+zdwdz(z2+ν2)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(z)=C1Iν(z)+C2Kν(z)

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

Kν(z)=π/2sin(νπ)(Iν(z)Iν(z))

Here Iν(z) are the modified Bessel functions of the first kind:

Iν(z)=(z/2)νπΓ(ν+1/2)0πezcos(t)sin(t)2νdt

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.

See Also

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