# Documentation

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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) ```

## Description

`H = besselh(nu,K,Z)` computes the Hankel function ${H}_{\nu }^{\left(K\right)}\left(z\right)$ 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 ${H}_{\nu }{}^{\left(K\right)}\left(z\right)$ by `exp(-i*Z)` if `K` = 1, and by `exp(+i*Z)` if `K` = 2.

## Examples

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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 [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```

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### Bessel’s Equation

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. 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

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

The relationship between the Hankel and Bessel functions is

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

where Jν(z) is `besselj`, and Yν(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.