It can also be the case that no one knows how to express the answer
in closed form in terms of known functions, and so it is not a problem
of computers, or of particular programs, but of current knowledge.
It may also be the case that someone knows an alternative to the
question, but the answer is too clumsy to present.
Under some restrictions on w, this integral is related to the
Ei (exponential integral), but with other assumptions, it
might not be expressible this way,
according to Mathematica 5.1. A numerical solution for
particular w might be more helpful.
Here's what mathematica says.
If[(Im[w] > 0 && Inequality[0, Less, Re[w], LessEqual,
Sqrt[2]*Im[w]]) || (Im[w] > 0 &&
Inequality[-(Sqrt[2]*Im[w]), LessEqual, Re[w],
Less, 0]) || (Im[w] < 0 && Inequality[0, Less,
Re[w], LessEqual, -(Sqrt[2]*Im[w])]) ||
(Im[w] < 0 && Inequality[Sqrt[2]*Im[w], LessEqual,
Re[w], Less, 0]) || (Im[w] != 0 && Re[w] == 0),
((-I/2)*(-(E^(2*Sqrt[2]*w)*ExpIntegralEi[
-((-I + Sqrt[2])*w)]) - ExpIntegralEi[
(-I + Sqrt[2])*w] + E^(2*Sqrt[2]*w)*
ExpIntegralEi[-((I + Sqrt[2])*w)] +
ExpIntegralEi[(I + Sqrt[2])*w]))/
(Sqrt[2]*E^(Sqrt[2]*w)),
Integrate[1/(E^(I*t*w)*(2 + t^2)), {t, -1, 1},
Assumptions -> !((Im[w] > 0 && Inequality[0, Less,
Re[w], LessEqual, Sqrt[2]*Im[w]]) ||
(Im[w] > 0 && Inequality[-(Sqrt[2]*Im[w]),
LessEqual, Re[w], Less, 0]) ||
(Im[w] < 0 && Inequality[0, Less, Re[w],
LessEqual, -(Sqrt[2]*Im[w])]) ||
(Im[w] < 0 && Inequality[Sqrt[2]*Im[w],
LessEqual, Re[w], Less, 0]) ||
(Im[w] != 0 && Re[w] == 0))]]
The commercial version of Macsyma gives this...
%i * (((%e)^( - sqrt(2) * w) * (sqrt(2) * exp_int((sqrt(2) - %i) * w) -
sqrt(2) * (%e)^(2 * sqrt(2) * w) * exp_int(( - %i - sqrt(2)) * w))/4)
- ((%e)^( - sqrt(2) * w) * (sqrt(2) * exp_int((%i + sqrt(2)) * w) - sqrt(2)
* (%e)^(2 * sqrt(2) * w) * exp_int((%i - sqrt(2)) * w))/4))
----- Original Message -----
From: <der_wachtmeister at freenet.de>
To: <maxima at math.utexas.edu>
Sent: Thursday, January 05, 2006 1:48 PM
Subject: Why do some integrations not work?
> Why can Maxima solve this:
>
> (%i3) integrate ( %e ^ ( -%i * w * t ), t, -1, 1 );
>
> and this not:
>
> (%i5) integrate ( %e ^ ( -%i * w * t ) / ( t ^ 2 + 2 ), t, -1, 1 );
>
> ? Is this a problem of Maxima (i.e. could Mathematica or Maple solve it)
> or of CAS in general?
>
> Phil
>
>
>