Do not know too much about that (and the question is a bit unclear).

Besides the certainly existing standard literature 2 links to NAG articles

http://www.nag.co.uk/doc/TechRep/Pdf/tr2_00.pdf and
http://www.nag.co.uk/Numeric/CL/financial/GLFEN27.pdf (view at the problem as a multi asset option and use quasi-random numbers)

20 items is quite high, so one way would be to put them into clusters, may be a PCA (principal component analysis) helps.

The only direct and brute way that I am aware of is for 'margin calculations', but that's on daily runs only (i.e. over-night risk)

The practical problem may be deeper: the pdf for the items usually would not be the same over livetime (except working with historical data, while a better approach might be to work with derived data). The more serious part is: the correlations are not constant and from my vague knowledge one should search for "dispersion trades" (i.e. a major risk comes from there).

For Maple I have seen some 'classical' stuff, but can not remember and just do not have Maple here to look up the Toolbox

I understand your complains. But would not agree in all your points.

It is certainly not true that the system ignores the assumptions of
the fundamental theorem. More likely it is a bug (may even be some
coding bug) and not a fundamental fault (i.e. ignoring).

In some cases Maple returns unevaluated, because it can not decide
whether the FT can be properly applied (have no example at hand now).

The other thing: automatical numerical cross checks. From time to
time this also is brought up at sci.math.symbolic - mainly there are
2 reasons why not to do: it will slow down the system (may be needed
in many intermediate steps) and in certain cases it will be difficult
to judge (say numerical almost impossible to achieve results or the
contrary (there is an article by Fateman(?) concerning the latter)).

The integral itself is far from being easy (remind that MMA does not
give an indefinite answer) and may be seen as ln((1+p(x)^2))/sqrt(1-x^2),
p(x)=1+x^2, to be integrated over the unit interval, by x=cos(t).

For other troublesome tasks you may for example search for Zacharias
"Some intriguing definite integrals" (1999).

So your frustration is understandable, but you picked up a peculiar
problem, I would not look at a CAS as a universal solver ... and
hope that encourages you again, at least a bit.

http://www.wias-berlin.de/main/publications/wias-publ/run.cgi?template=abstract&type=Preprint&year=1999&number=493

the jumps where to cut and take the directed limits for the anti-derivate

1/2*Pi-1/2*arccos(5^(1/2)-2^(1/2)) and 1/2*Pi+1/2*arccos(5^(1/2)-2^(1/2))
(i used 'discont', stared at the solution and with it, searching a Real one)

however M11 needs evalf to get the limits

seems MMA does not give the anti-derivative (online version)

PS: the new editor is a mess for line breaks, deleting and back space
only usable if typing first in a usual editor, then do copy+paste

Concerning your question whether it may be useful for learning I would say "No, learn the basics the classical way", even if I am not used to all the stuff

  Int( ln(2+2*cos(t)^2+cos(t)^4), t = 0 .. Pi);
  S:=simplify(combine(value(%)));

       / -----
       |  \
  S := |   )
       |  /
       | -----
       \_R1 = %1

                                                             \
        /       _R1 - 1          _R1 + 1            _R1 + 1 \|
        |-dilog(-------) + Pi ln(-------) I + dilog(-------)|| I
        \         _R1              _R1                _R1   /|
                                                             |
                                                             /

                            2
         + Pi ln(5) + 2 I Pi

                    4        2        6         8
  %1 := RootOf(54 _Z  + 12 _Z  + 12 _Z  + 1 + _Z )

  S1:=select(has,S,sum);
  S2:=S-S1;

Now evaluating S1 ( = the sum from above) shows the error, since the integral is real.

personally i would not like to get notified for each thread i ever posted, but only for those i subscribed

and i guess the feature will not scan posts beyond it

thus your posts without subscribtions ( = while the feature was not present) should not come up

no problem, that's what testing is good for - thx for bringing it back!

i find it just after the first post of the thread

however i do not see, how to subscribe a thread (for example this one) for notification on new contributions

Thx ... either wheat beer or burgundy or homoepathic stuff, pills should be the least ...

my feeling (after some looking around) says, one should try to look at or use something like

GAMMA(z)*Zeta(z)=Int(t^(z-1)/(exp(t)-1),t=0 .. infinity);

                                   infinity
                                  /           (z - 1)
                                 |           t
             GAMMA(z) Zeta(z) =  |          ---------- dt
                                 |          exp(t) - 1
                                /
                                  0
with 1 < Re(z)

Thx ... either wheat beer or burgundy or homoepathic stuff, pills should be the least ...

my feeling (after some looking around) says, one should try to look at or use something like

GAMMA(z)*Zeta(z)=Int(t^(z-1)/(exp(t)-1),t=0 .. infinity);

                                   infinity
                                  /           (z - 1)
                                 |           t
             GAMMA(z) Zeta(z) =  |          ---------- dt
                                 |          exp(t) - 1
                                /
                                  0
with 1 < Re(z)

arrgghh ... my stomach ... :-)

Ok, will try to have a look, hope I will no get crazy ... any accessible reference for that woodoo?

Edited to add: I think one actually needs that the real part of beta is negative for both the steps.
For example the sum for U up to factors is the series for the zeta function Zeta(0,z,2+alpha),
z = -beta+1 and the extension needs 1 < Re(z),  Zeta(0,z,2+alpha) = Zeta(z) for alpha=0.

So you compute by extending without the very restriction.

May be one can either modify the double regularization. Or tries to work without convergence
(and an artifical variable for the sum, i.e. formal power series for U and V). Or ... I have no good
idea, at least Zeta( -beta + 1) ---> Zeta(0) = -1/2.

arrgghh ... my stomach ... :-)

Ok, will try to have a look, hope I will no get crazy ... any accessible reference for that woodoo?

Edited to add: I think one actually needs that the real part of beta is negative for both the steps.
For example the sum for U up to factors is the series for the zeta function Zeta(0,z,2+alpha),
z = -beta+1 and the extension needs 1 < Re(z),  Zeta(0,z,2+alpha) = Zeta(z) for alpha=0.

So you compute by extending without the very restriction.

May be one can either modify the double regularization. Or tries to work without convergence
(and an artifical variable for the sum, i.e. formal power series for U and V). Or ... I have no good
idea, at least Zeta( -beta + 1) ---> Zeta(0) = -1/2.

finally you set beta = 1, but to get U and V you assume beta < 0

finally you set beta = 1, but to get U and V you assume beta < 0

Thanks for the hint, however I just had in mind some routines to produce Gauss quadrature rules (for given weight functions for example), there are some standard ways (sketched in the cited document) - certainly easier then the theory around Koepf's work.