besselh

Bessel function of third kind (Hankel function) for symbolic expressions

Syntax

H = besselh(nu,K,z)

H = besselh(nu,z)

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

Description

H = besselh(nu,K,z) computes the Hankel function $$H_{\nu }^{(K)}(z)$$, where K = 1 or 2, for each element of the complex array z. The output H has the symbolic data type if any input argument is symbolic. See Bessel’s Equation.

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

H = besselh(nu,K,z,1) scales $$H_{\nu }^{(K)}(z)$$ by exp(-i*z) if K = 1, and by exp(+i*z) if K = 2.

Examples

Compute Hankel Function

Specify the Hankel function for a symbolic variable.

syms z
H = besselh(3/2,1,z)

H = 
$$-\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{z\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\left(1+\frac{\mathrm{i}}{z}\right)}{\sqrt{z}\hspace{0.17em}\sqrt{\pi }}$$

Evaluate the function symbolically and numerically at the point z = 1 + 2i.

Hval = subs(H,z,1+2i)

Hval = 
$$\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{-2+\mathrm{i}}\hspace{0.17em}\left(-\frac{7}{5}-\frac{1}{5}\hspace{0.17em}\mathrm{i}\right)}{\sqrt{1+2\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\sqrt{\pi }}$$

vpa(Hval)

ans = $$-0.084953341280586443678471523210602-0.056674847869835575940327724800155\hspace{0.17em}\mathrm{i}$$

Specify the function without the second argument, K = 1.

H2 = besselh(3/2,z)

H2 = 
$$-\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{z\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\left(1+\frac{\mathrm{i}}{z}\right)}{\sqrt{z}\hspace{0.17em}\sqrt{\pi }}$$

Notice that the functions H and H2 are identical.

Scale the function by $$e^{-iz}$$ by using the four-argument syntax.

Hnew = besselh(3/2,1,z,1)

Hnew = 
$$-\frac{\sqrt{2}\hspace{0.17em}\left(1+\frac{\mathrm{i}}{z}\right)}{\sqrt{z}\hspace{0.17em}\sqrt{\pi }}$$

Find the derivative of H.

diffH = diff(H)

diffH = 
$$\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{z\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{z^{5/2}\hspace{0.17em}\sqrt{\pi }}-\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{z\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\left(1+\frac{\mathrm{i}}{z}\right)\hspace{0.17em}\mathrm{i}}{\sqrt{z}\hspace{0.17em}\sqrt{\pi }}+\frac{\sqrt{2}\hspace{0.17em}{\mathrm{e}}^{z\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\left(1+\frac{\mathrm{i}}{z}\right)}{2\hspace{0.17em}{z}^{3/2}\hspace{0.17em}\sqrt{\pi }}$$

Input Arguments

nu — Hankel function order

symbolic array | double array

Hankel function order, specified as a symbolic array or double array. 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 size.

Example: nu = 3*sym(pi)/2

K — Kind of Hankel function

symbolic 1 or 2 | double 1 or 2

Kind of Hankel function, specified as a symbolic or double 1 or 2. K identifies the sign of the added Bessel function Y:

$$\begin{array}{l}{H}_{\nu }^{(1)}(z)=J_{\nu }(z)+i{Y}_{\nu }(z)\\ H_{\nu }^{(2)}(z)=J_{\nu }(z)-i{Y}_{\nu }(z).\end{array}$$

Example: K = sym(2)

z — Hankel function argument

symbolic array | double array

Hankel function argument, specified as a symbolic array or double array. 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 size.

Example: z = sym(1+1i)

More About

Bessel’s Equation

The differential equation

$$z^2\frac{d^{2}w}{d{z}^2}+z\frac{dw}{dz}+\left(z^2-{\nu }^2\right)w=0,$$

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

J_ν(z) and J_–ν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν. Y_ν(z) is a second solution of Bessel's equation—linearly independent of J_ν(z)—defined by

$$Y_{\nu }(z)=\frac{J_{\nu }(z)\cos (\nu \pi )-J_{-\nu }(z)}{\sin (\nu \pi )}.$$

The relationship between the Hankel and Bessel functions is

$$\begin{array}{l}{H}_{\nu }^{(1)}(z)=J_{\nu }(z)+i{Y}_{\nu }(z)\\ H_{\nu }^{(2)}(z)=J_{\nu }(z)-i{Y}_{\nu }(z).\end{array}$$

Here, J_ν(z) is besselj, and Y_ν(z) is bessely.

References

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

Version History

Introduced in R2018b