Hankel function, mathematical definition
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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.
3 Comments
Stephen23
on 20 Dec 2021
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.
Accepted Answer
David Goodmanson
on 12 Dec 2021
Edited: David Goodmanson
on 12 Dec 2021
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.
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