MATLAB Function Reference |

Bessel function of the third kind (Hankel function)

**Syntax**

**Definitions**

where is a nonnegative constant, is called *Bessel's equation*, and its solutions are known as *Bessel functions*. and form a fundamental set of solutions of Bessel's equation for noninteger . is a second solution of Bessel's equation - linearly independent of - defined by

The relationship between the Hankel and Bessel functions is

where is `besselj`

, and is `bessely`

.

**Description**

```
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. If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.

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

scales by `exp(-i*Z)`

if `K`

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

if `K`

= 2.

```
[H,ierr] = besselh(...)
```

also returns completion flags in an array the same size as `H`

.

**Examples**

This example generates the contour plots of the modulus and phase of the Hankel function shown on page 359 of [1] Abramowitz and Stegun, *Handbook of Mathematical Functions*.

It first generates the modulus contour plot

[X,Y] = meshgrid(-4:0.025:2,-1.5:0.025:1.5); H = besselh(0,1,X+i*Y); contour(X,Y,abs(H),0:0.2:3.2), hold on

then adds the contour plot of the phase of the same function.

**See Also**

`besselj`

, `bessely`

, `besseli`

, `besselk`

**References**

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

beep | besseli |