Re: NEC-LIST: Symbolic evaluation of integrals

Regarding your integral...
I am packing to move, so I don't have my books available. However the following
mehtod should give an analytical answer (in terms of a sum Besel functions,
unfortunately)

Make the substitution r = cosh[t] = (exp[t] + exp[-t])/2

The integral F[r,k] can then be written as

F[r,k] = Integral[ {exp[-ik cosh[t]] / cosh[t]} dt]

Then take the partial derivative to get

d F/ d k = -i Integral[ exp[-ik cosh[t]] dt]

remember then the formula

exp[i z cos(w)] = Sum[ n=-infinity to infinity {i**n Jsub-n[z] exp[i n w]}]

Let w = - it, and z = -k giving

exp[-i k cosh[t]] = Sum[n=-infinity to infinity {i**n Jsub-n[-k] exp[nt]} ]

You might think this sum doesn't converge for large n, due to the exp[nt]. I
think that the Besell fn will decay fast enough for it to converge, but my
Abbromowitz-Stegun is packed, so ...

Now, the integral becomes an integral over t for exp[nt], which is easy. Then,
you must integrate over k to undo the partial derivative. That is, you must
integrate Jsub-n[-k] over k. I believe there is a recursion relation that gives
that answer in terms of two Bessel fns. So, there is an answer for your problem.
It will be an infinite sum of Bessel fns.

                Let me know how it works,

Francis Canning

Bibby wrote:

> I have what appears to be, on the face of it, a rather simple integral that
> I would like to have an analytical solution for. I have not been able to
> derive one or find one in the many tabulations that I have available.
>
> A possible solution is the use of symbolic evaluation. I would be
> grateful to hear from anyone who has access to one of these packages (Axiom,
> Derive, Macsyma, MAPLE, Mathematica, Reduce and any others) that would run
> the test for me. Again, the integrand is deceptively simple, so it won't
> take long to type in.
>
> Thanks,
>
> Malcolm M Bibby
> --
> The NEC-List mailing list <nec-list_at_gweep.ca>
> http://www.gweep.ca/mailman/listinfo.cgi/nec-list

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