MATLAB Function Reference    

Bessel function of the third kind (Hankel function)



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.


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.

besselh successfully computed the Hankel function for this element.
Illegal arguments.
Overflow. Returns Inf.
Some loss of accuracy in argument reduction.
Unacceptable loss of accuracy, Z or nu too large.
No convergence. Returns NaN.


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

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

See Also

besselj, bessely, besseli, besselk


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

  beep besseli