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

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

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

$${Y}_{\nu}(z)=\frac{{J}_{\nu}(z)\mathrm{cos}(\nu \pi )-{J}_{-\nu}(z)}{\mathrm{sin}(\nu \pi )}.$$

The relationship between the Hankel and Bessel functions is

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

where *J*_{ν}(*z*) is `besselj`

,
and *Y*_{ν}(*z*) is `bessely`

.

`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`

.
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 $${H}_{\nu}{}^{(K)}(z)$$ by `exp(-i*Z)`

if `K`

=
1, and by `exp(+i*Z)`

if `K`

= 2.

[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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