Re: help with an integral which looks similar to elliptic integrals.
- From: "len" <i.e.linington@xxxxxxxxxxxx>
- Date: 3 Jul 2006 16:36:49 -0700
Hi Gene,
Sorry, I made a slip in the last email.
It should have read
**A IS REAL AND -VE, B IS IMAGINARY AND C HAS BOTH REAL AND IMAGINARYI 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,
PARTS.**
-sorry about that.
Actually, this *is* of genus one, and you can find an answer in terms
of elliptic integrals.
Try specializing A, B, C, D, n and m (why both?) to particular values
and feeding it to Maple or Mathematicia.
but what about this?
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'm not aware of a way to specify the contour in Mathematica other than
to go back to the original variable x, i.e.
\int_{0}^{2pi} (exp(q*ix) / (\sqrt{A*exp(4ix) + B*exp(3ix) + C*exp(2ix)
- B*exp(ix) +A}) dx
with q integer and A, B, C as above.
I tried putting explicit values in for A, B, C and q, i.e.
Integrate[
Exp[6*I*x]/(Sqrt(-1*Exp[4*I*x] - 2*I*Exp[3*I*x] + 2(1 + I)*Exp[2*I*x]
+
2*I*Exp[I*x] - 1)), {x, 0, 2*Pi}]
but Mathematica simply returns the input as output without evaluating
the integral.
Any thoughts?
Thanks,
Len
.
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