From: Walter Roberson on
Walter Roberson wrote:
> Nathan Zhang wrote:
>
>> Even the value of "nu" is from 0 to +infinity as integer, how to find
>> the sum??
>
> You cannot. The besselh documentation indicates that besselh is not
> defined when nu is an integer. If you examine the definition of besselh,
> you will see there is a sin(nu*Pi) in the denominator, and that is
> going to be 0 for every integer nu .

I have sent a service request to Mathworks on this matter. Mathworks documents
the individual components of the equations as being equivalent to besselj and
bessely, and if you examine either of those you will see that they imply that
integral v is common for their inputs. The documentation may be incorrect in
requiring non-negative non-integral v (nu).

Whether the documentation is correct or not, it is the case that there is no
easily calculable closed-form function for the summation you are requesting.
From: Walter Roberson on
Walter Roberson wrote:

> My symbolic package is telling me that
> simplify(convert(Sum(f,n=-10..10),Int)) is 0, but with numeric
> substitution, I think it is mistaken.

With some assistance from other users, I have proven that the above is a
bug in the symbolic package I am using, that the conversion to integral
form should _not_ result in 0 for the summation.

Unfortunately, the formula that I am able to come up with do not appear
to have any closed form.

It appears to me that as you add more and more components (extend the
summation further and further towards infinity) that you will more and
more closely approximate something akin to a trapazoid or possibly
parallelagram wave. The exp() part you have in the numerator would, by
itself, have a limit that was a square wave (I think), but the besselh
in the denominator adds a slightly different phase correction to each of
the sine waves. I could be misinterpreting.