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 ....