Hello Bart,
g(x) := exp( -( (a-b*cos(x))^2 + (c-b*sin(x))^2 ) );
integrate((g(x))^2, x, 0, 2*%pi);
A combination of radcan, and trigsimp reduces this integral to the
following:
integrate((%e)^(4 * b * c * sin(x) + 4 * a * b * cos(x) - 2 * c^2 - 2 *
b^2 - 2 * a^2),x,0,2 * %pi)
The constant -2*(a^2+b^2+c^2) can be factored out.
There is a trig identity which converts 4*a*b*(sin(x)+cos(x)) to
sqrt(2)*4*a*b*cos(x-atan2(a,b)).
Then the integrand can be expanded into I0(4*sqrt(2)*a*b)+ a sum
containing terms multiplied by cos(n*(x+atan2(a,b))
which when integrated between 0 and 2*pi give 0.
So I think this integral is exp(-2*(a^2+b^2+c^2)*2*pi*I0(4*sqrt(2)*a*b).
However, I don't think the Maxima integrator knows how to apply either
of the last two identities, but you could write pattern matching rules.
Dan
Bart Vandewoestyne wrote:
> Hello list,
>
> Today, a collegue of me presented me the following integral:
>
> g(x) := exp( -( (a-b*cos(x))^2 + (c-b*sin(x))^2 ) );
> integrate((g(x))^2, x, 0, 2*%pi);
>
> He claims he was able to solve this integral in terms of Bessel functions, but
> for practical reasons, he would like to be able to obtain the value of the
> integral with a CAS. He has tried Maple, but he wasn't able to solve it.
>
> I wanted to try it with Maxima... the above 2 lines of code are my attempt,
> but apparently, Maxima can't handle it this way.
>
> Are there any (Maxima) tricks I can use in order to obtain a solution with
> Maxima?
>
> Thanks,
> Bart
>
>