MATLAB Function Reference
besselh

Bessel function of the third kind (Hankel function)

Syntax

• ```H = besselh(nu,K,Z)
H = besselh(nu,Z)
H = besselh(nu,K,Z,1)
[H,ierr] = besselh(...)
```

Definitions

The differential equation

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,Z) ``` uses `K` = 1.

```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`.

 ierr Description `0` `besselh` successfully computed the Hankel function for this element. `1` Illegal arguments. `2` Overflow. Returns `Inf`. `3` Some loss of accuracy in argument reduction. `4` Unacceptable loss of accuracy, `Z` or `nu` too large. `5` No convergence. Returns `NaN`.

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.

• ```contour(X,Y,(180/pi)*angle(H),-180:10:180); hold off

```

`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