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besselh

Bessel function of third kind (Hankel function)

Syntax

H = besselh(nu,K,Z)
H = besselh(nu,Z)
H = besselh(nu,K,Z,1)

Definitions

The differential equation

z2d2ydz2+zdydz+(z2ν2)y=0,

where ν is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions. Jν(z) and Jν(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

Yν(z)=Jν(z)cos(νπ)Jν(z)sin(νπ).

The relationship between the Hankel and Bessel functions is

Hν(1)(z)=Jν(z)+iYν(z)Hν(2)(z)=Jν(z)iYν(z),

where Jν(z) is besselj, and Yν(z) is bessely.

Description

H = besselh(nu,K,Z) computes the Hankel function Hν(K)(z) 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.

H = besselh(nu,Z) uses K = 1.

H = besselh(nu,K,Z,1) scales Hν(K)(z) by exp(-i*Z) if K = 1, and by exp(+i*Z) if K = 2.

Examples

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Modulus and Phase of Hankel Function

This example generates the contour plots of the modulus and phase of the Hankel function $H_0^{(1)}(z)$ shown on page 359 of Abramowitz and Stegun, Handbook of Mathematical Functions [1].

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

References

[1] Abramowitz, M., and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965.

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