I think I found a bug in MapleV.4(00f).
evaluating
a:=int(exp(-x)/(1+x^2),x=0..infinity);
I get
a:=(I/2) (exp(I) Ei(1,I) - exp(-I) Ei(1,-I))
with
evalf(a) = 0.6214496240....
On the contrary,
evalf(evalc(a))= -1.0759... + I 2.643559
which is obviously wrong.
I suspect the bug is in the evalc() command.
In fact,
MapleV.3
evalc(Ei(1,I)) = - Ci(1) + I (Si(1) - 1/2 Pi) evalc(Ei(1,-I)) = - Ci(1) - I (Si(1) - 1/2 Pi)
MapleV.4
evalc(Ei(1,I)) = - Ci(1) + I (Si(1) - 1/2 Pi) evalc(Ei(1,-I)) = - Ci(1) - I (Si(1) + 3/2 Pi) ^^^^^^^^
The bug is removed with Maple V Release 5. (U. Klein)
Continuing on this theme of Lucca Ciotti (sorry to say it, but Maple is a goldmine if it comes down to errors in branch cuts of complex functions):
The integral of a real function like: log(sin(x)) is of course determined up to a constant. If
the integral happens to be explicitly available, this constant may be appear to be complex,
and I can understand that if one of the occurring functions is multivalued (and has branches,
branch cuts, etc) this constant may jump from one value to another along the
x-interval. (This is why it is always tricky to accept a result from int(f(x),x=a..b)
straightaway.)
However, the imaginary part of the integral of log(sin(x)) is not piecewise constant on
0..Pi!
Between 0 and Pi/2 it is constant, but it decays linearly between Pi/2 and Pi.
(Then along Pi..2 Pi it also grows linearly, but this is how it should, because
sin(x)<0, and log(sin(x)) an imaginary constant.)
So there is something more fishy going on ....