Re: A (slightly more) complicated integral

From: G. A. Edgar (edgar_at_math.ohio-state.edu.invalid)
Date: 06/18/04 

Date: Fri, 18 Jun 2004 09:57:47 -0400

In article <40D2CFE9.2070005@leeds.ac.uk>, Enrique Aguado
<l.e.aguado@leeds.ac.uk> wrote:

> Ok, so the previous integral ended up being a piece of cake,
> however this one cannot be done in the same way (and it ends up that
> this one is the one I really want, isn't it always the same?)
>
> It looks like this:
>
> Int[Int[E^(a Cos[x]+ b Cos[y]+ k Cos[x - y]) {y, -Pi, Pi}],{x, -Pi, Pi}]
>
> Any suggestions anyone?
>
>
> Many many thanks in advance
>
> Enrique
>

Well, not a solution, but I do have these...

Do these in order (I'm using Maple)...

> Int(exp(r*cos(x)),x=-Pi..Pi) = 2*Pi*BesselI(0,r);
            /Pi
           |
           | exp(r cos(x)) dx = 2 Pi BesselI(0, r)
           |
          /-Pi

> Int(exp(r*cos(x-T)),x=-Pi..Pi) = 2*Pi*BesselI(0,r);
          /Pi
         |
         | exp(r cos(x - T)) dx = 2 Pi BesselI(0, r)
         |
        /-Pi

> Int(exp(A*cos(x)+B*sin(x)+C),x=-Pi..Pi) = 2*Pi*exp(C)*BesselI(0,sqrt(A^2+B^2));
        /Pi
       |
       | exp(A cos(x) + B sin(x) + C) dx =
       |
      /-Pi

                      / (1/2)\
                      | / 2 2\ |
   2 Pi exp(C) BesselI\0, \A + B / /

> expand(a*(cos(x)+cos(y))+b*cos(x-y));
    a cos(x) + a cos(y) + b cos(x) cos(y) + b sin(x) sin(y)

  /Pi
 |
 | exp((a + b cos(y)) cos(x) + b sin(x) sin(y) + a cos(y)) dx =
 |
/-Pi

                          / (1/2)\
                          | / 2 2 2\ |
2 Pi exp(a cos(y)) BesselI\0, \(a + b cos(y)) + b sin(y) / /

So your double integral is:

> 2*Pi*int(exp(a*cos(y))*BesselI(0, (a^2+2*b*cos(y)*a+b^2)^(1/2)), y)

I didn't find this, but there may be some hope, because G&R does have
it without the exp...

         /Pi / (1/2)\
        | | / 2 2\ |
        | BesselI\0, \a + 2 b cos(y) a + b / / dy =
        |
       /-Pi

                2 Pi BesselI(0, a) BesselI(0, b) 

-- 
G. A. Edgar                               http://www.math.ohio-state.edu/~edgar/