Re: help with an integral which looks similar to elliptic integrals.

Hi Gene, thanks for the help!

The reason that I want to do the integral is for a research problem in
quantum optics. I'm trying to calculate the decay rate of an atom into
an engineered cavity and in the model we're using, the decay rate boils
down to a sum of integrals of the given form. It would be much nicer if
we could solve the integral and give an analytic solution rather than
leave it as an integral.

I tried the substitution you suggested and end up with an integral of
the form

\int (z^{n+m}) / (\sqrt{A*z^4 + B*z^3 + C*z^2 - B*z +A}) dz

where n and m are integer, and A, B, C are all real. (Can guarantee
that A and B are positive, but C could be +ve or -ve.)

However, the integral is now a contour integral, around the unit
circle. It looks to me that the branch cuts of this function will make
this impossible by contour integration. I've had a brief look on the
internet about algebraic functions and compact Riemann surfaces and it
looks fairly complicated. I'm unfamiliar with this stuff at the moment.


Could you offer any advice as to what you think the chances of an
analytic solution might be? Any help you can give would be greatly
received!!

Many thanks,
Len



Gene Ward Smith wrote:
len wrote:

The form is reminiscent of elliptic integrals, but it doesn't seem
quite close enough to
evaluate easily...

If you substitute z = exp(ix), then the integral becomes algebraic.
Hiwever, in general the genus is greater than 1, and you don't get
elliptic integrals. Instead, you've wandered into the world of
algebraic functions and compact Riemann surfaces. What do you want the
integral for? You may want to avoid getting into deeper waters here.

.