Modified Bessel function of second kind
K = besselk(nu,Z)
K = besselk(nu,Z,1)
The differential equation
where ν is a real constant, is called the modified Bessel's equation, and its solutions are known as modified Bessel functions.
A solution Kν(z) of the second kind can be expressed as:
where Iν(z) and I–ν(z) form a fundamental set of solutions of the modified Bessel's equation,
and Γ(a) is the gamma function. Kν(z) is independent of Iν(z).
Iν(z) can be computed using besseli.
K = besselk(nu,Z) computes the modified Bessel function of the second kind, Kν(z), for each element of the array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive.
If nu and Z are arrays of the same size, the result is also that size. If either input is a scalar, it is expanded to the other input's size.
Create a column vector of domain values.
z = (0:0.2:1)';
Calculate the function values using besselk with nu = 1.
format long besselk(1,z)
ans = Inf 4.775972543220472 2.184354424732687 1.302834939763502 0.861781634472180 0.601907230197235
Define the domain.
X = 0:0.01:5;
Calculate the first five modified Bessel functions of the second kind.
K = zeros(5,501); for i=0:4 K(i+1,:) = besselk(i,X); end
Plot the results.
plot(X,K,'LineWidth',1.5) axis([0 5 0 8]) grid on; legend('K_0','K_1','K_2','K_3','K_4','Location','Best') title('Modified Bessel Functions of the Second Kind for v = 0,1,2,3,4') xlabel('X') ylabel('K_v(X)')
 Abramowitz, M., and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sections 9.1.1, 9.1.89, and 9.12, formulas 9.1.10 and 9.2.5.
 Carrier, Krook, and Pearson, Functions of a Complex Variable: Theory and Technique, Hod Books, 1983, section 5.5.
 Amos, D.E., "A Subroutine Package for Bessel Functions of a Complex Argument and Nonnegative Order," Sandia National Laboratory Report, SAND85-1018, May, 1985.
 Amos, D.E., "A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order," Trans. Math. Software, 1986.