Hankel function, mathematical definition

Hello everyone,

I'm wonder about the besselh(.) function.

The definition given is,

H = besselh(nu,K,Z,scale) specifies whether to scale the Hankel function to avoid overflow or loss of accuracy. If scale is 1, then Hankel functions of the first kind H(1)ν(z) are scaled by e−iZ, and Hankel functions of the second kind H(2)ν(z) are scaled by e+iZ.

But I found that (in eq. 12.140-2, Weber & Arfken, 2003)

Hankel first kind:

Hankel second kind:

That mean H(1)ν(z) correspond to and H(2)ν(z) correspond to ? and why that is inverted so ?

Thank you.

[David Goodmanson]

Hi Kevin,

Once a question has been asked and answered, or even commented on in a significant way, there is a policy on this site (maybe unwritten) that questions should not be deleted. One reason is that deleting questions can take away potentially useful information for users of the site, which is of course searchable. Anyway, this question has been here for more than a week without objection and the answer about normalization addresses Matlab documentation that probably could have been stated better. So I think that the question is appropriate for Matlab Answers. On these grounds, do you concur?

[Stephen23]

https://web.archive.org/web/20211220041046/https://www.mathworks.com/matlabcentral/answers/1609065-hankel-function-mathematical-definition

Hankel function, mathematical definition

Hello everyone,

I'm wonder about the besselh(.) function.

The definition given is,

H = besselh(nu,K,Z,scale) specifies whether to scale the Hankel function to avoid overflow or loss of accuracy. If scale is 1, then Hankel functions of the first kind H(1)ν(z) are scaled by e−iZ, and Hankel functions of the second kind H(2)ν(z) are scaled by e+iZ.

But I found that (in eq. 12.140-2, Weber & Arfken, 2003)

Hankel first kind:

Hankel second kind:

That mean H(1)ν(z) correspond to and H(2)ν(z) correspond to ? and why that is inverted so ?

Thank you.

[Rik]

Regarding your flag, why should this question be deleted?

Answer 1

[David Goodmanson]

Hi Kevin,

The hankel functions h that you cited are spherical hankel functions, which have half-integer order and are related to the regular hankel function H by

h(n,1,z) = const/sqrt(z)*H(n+1/2,1,z)       % first kind
h(n,2,z) = const/sqrt(z)*H(n+1/2,2,z)       % second kind
where
H(m,1,z) = besselh(m,1,z) 
H(m,2,z) = besselh(m,2,z)

To the best of my knowledge (I have 2019b), spherical bessel functions still are not a part of core Matlab.

Those details do not change the basic question about normalization. For large z,

besselh(m,1,z) --> const/sqrt(z)*exp(i*z)    as |z| --> inf
besselh(m,2,z) --> const/sqrt(z)*exp(-i*z)   as |z| --> inf

so the first kind goes like exp(i*z) and the second kind goes like exp(-i*z) as you said.

For larger but not overly large z, the factor in front is a slowly varying function that goes over to const/sqrt(z) in the limit.

Including scaling just means that the bessel function of the first kind is multiplied by exp(-i*z) to make the known exponential factor go away, leaving the slowly varying function. Similarly for the second kind.

[Kevin ROUARD]

Hi David,

thank you for the answer. Effectively, I speak about spherical Hankel functions. Thanks for the scaling explanation.