I have seen an equation about Hankle function,but I don't know why?
$(\frac{\partial^2 }{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\beta )H^{(1)}_{0}(\sqrt{\beta} r)=\frac{4i\delta(r)}{2\pi r}$
what's more, is there a similar equation for the second kind modified bessel function. thanks.
http://math.stackexchange.com/questions/1607624/an-equation-about-hankle-function
Dear Liu Yong,
Can you send your equation in the form of pdf or other file. I can't read the text correctly.
Sincerely,
P. Malits
Dear Liu Yong,
Can you send your equation in the form of pdf or other file. I can't read the text correctly.
Sincerely,
P. Malits
Hello Liu,
You check the validity of the statement, first the real part, which is true by inspection; then the imaginary part, which is a little tricky, but you need to calculate the residue of the result, where the result is derived from applying all the operators on the Hankel function.
After simplifying and checking the result, myself, I think the value on the right-hand side of the equality is wrong, I do believe it is twice the value it should be...
And, for the modified Bessel functions of either the first or second kind, they both are relatable to the Bessel function of the first kind, hence, you need to find the relations and modify the equation yourself, plus, you can use the same residue technique to modify the right-hand side of the equation... Done.
The reason this relation exists is simple, we know the Bessel functions are a solution to the radial equation, hence, one may modify as one pleases. The reason the right-hand side is not zero, it is because the second term of a Hankel function is indeterminate at the origin.
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