Bessel function of third kind (Hankel function)
H = besselh(nu,K,Z)
H = besselh(nu,Z)
H = besselh(nu,K,Z,1)
The differential equation
where ν is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions. Jν(z) and J–ν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν. Yν(z) is a second solution of Bessel's equation—linearly independent of Jν(z)—defined by
The relationship between the Hankel and Bessel functions is
where Jν(z) is besselj, and Yν(z) is bessely.
H = besselh(nu,K,Z) computes the Hankel function where K = 1 or 2, for each element of the complex array Z. If nu and Z are arrays of the same size, the result is also that size. If either input is a scalar, besselh expands it to the other input's size.
This example generates the contour plots of the modulus and phase of the Hankel function shown on page 359 of Abramowitz and Stegun, Handbook of Mathematical Functions .
Create a grid of values for the domain.
[X,Y] = meshgrid(-4:0.025:2,-1.5:0.025:1.5);
Calculate the Hankel function over this domain and generate the modulus contour plot.
H = besselh(0,1,X+1i*Y); contour(X,Y,abs(H),0:0.2:3.2), hold on
In the same figure, add the contour plot of the phase.
contour(X,Y,(180/pi)*angle(H),-180:10:180); hold off
 Abramowitz, M., and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965.