complicated integration

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complicated integration

Posted on 2008-07-27 17:28 By zher (8)

Categories:

How do I ...? with Maple

i have this integration:

int(q*exp((1-b)^2* q_z^2/lambda )/((1+b^2*q^2+b^2*q_z^2)*(q^2+q_z^2)),q_z=-infinity..infinity) then w.r.t q from (0......infinity)

the problem is how to get rid of float (infinity ) from results to be able continue the problem and make minimize for the results terms. by the way i used simpson method and making change variables q to (1-y) /y then the integration become q from 0 to 1 because numerical method does not deal with infinity ...

spam

Comment on 2008-07-27 17:41 from alec ( Maple Rating 4

1476)

I might help you if you didn't post that 6 times in the forums and 4 times in private messages.

Alec

Community Guidelines

Comment on 2008-07-27 23:20 from Tim Vrablik (

69)

Zher,

I suggest you read over the MaplePrimes Community Guidelines, more specifically point #5.

The 5 repeated posts have been removed.

ranges

Comment on 2008-07-27 23:46 from alec ( Maple Rating 4

1476)

I think, you made some mistakes in your equations (if I understand them right.).

Nevertheless. In the form they are written, the double integral is infinity for b=1 (because of the singularity at 0) and is positve and approaching 0 for lambda approaching -0 (if lambda is positive, the double integral is infinite. and for lambda=0 it is not defined) if b is not equal to 1. So your minimization problem doesn't have much sense unless you also state the ranges for b and lambda.

Alec

expectation value?

Comment on 2008-07-28 06:46 from Axel Vogt ( Maple Rating 4

948)

I think he wants to minimize an expectation value w.r.t. that q
and that the integral (up to factors) can be written as 

  Int( exp(-x^2) * f(x,q), x = -infinity .. infinity), 
  f(x,q)=q/polynom(x,q)

where that polynomial in x has 4 distinct complex roots.

Maple does not give the integral in closed form (even after using
(real) partial fractions), i.e. the special function(s) involved
are possibly not of known form. And even if it would, it is not
clear that if would lead to some usefull result for q_min.

And 0 = diff(theIntegral,q) looks not that nice to solve ...

May be VariationalCalculus can be used. But I guess, one needs
sound Math, not only Maple, for a usefull description of the
desired aim.

none

Comment on 2008-07-28 15:46 from Axel Vogt ( Maple Rating 4

948)

If the question is as I suppose (but that guy seems to be too lazy
to answer the implicite question), then I think there is not extremum
except at the boundaries.

E0:=Int(q*exp((1-b)^2* z^2/lambda )/((1+b^2*q^2+b^2*z^2)*(q^2+z^2)),
  z=-infinity..infinity);

E:=eval(E0, lambda=-alpha^2); # since 0 < lambda ===> infinite
E1:=PDETools[dchange](z = x/(1-b)*alpha, %, [x]) assuming 0 < alpha:

That depends on sign( b-1 ), but changes only the sign (use 'Flip').

S:=[b = 1/(s^(1/2)*beta), q = p^(1/2)*beta, alpha = beta-1/(s^(1/2))];

  E1 assuming b-1 < 0;
  eval(%, S);
  PS:=simplify(%) ;

                       infinity
                      /                    2   1/2
                     |             s exp(-x ) p
              PS :=  |          --------------------- dx
                     |                2    2
                    /           (p + x ) (x  + p + s)
                      -infinity

  plot3d(PS, s=0..2,p=20..1, axes=boxed);

This plot suggests, that the value is decreasing with increasing p
(but I am too lazy too for trying a proof), which should give that 
there is no finite minimum.

Proff. Alec I am sorry

Comment on 2008-07-28 15:48 from zher (8)

Proff. Alec

I am sorry if I made any trouble for you

sorry , i forget the term so

Comment on 2008-07-28 16:08 from zher (8)

sorry , i forget the term so the question is :

int ( q * exp((1-b)^2* q_z^2 / lambda[1] ) * exp((1-b)^2* q^2 / lambda[2]) / ((1+b^2*q^2+b^2*q_z^2)*(q^2+q_z^2)) , q_z= -infinity..infinity) then w.r.t q from (0......infinity)

after integration we want to minimize the results , the range is as (b=0..1), ( lambda[1]) =1....100000) , (lambda[2]) =1....100000

then

Comment on 2008-07-28 16:43 from Axel Vogt ( Maple Rating 4

948)

then it is to mimize the following w.r.t. variable p

exp(rho*p)*J1;


                             infinity
                            /                     2
                    1/2    |                exp(-x )
        exp(rho p) p    s  |          --------------------- dx
                           |            2                2
                          /           (x  + p + s) (p + x )
                            -infinity

where rho=(1-b)^2/ lambda[2], lambda = lambda[1] in the above &
using your inital question.

Then a minimum seems to exist, probably you have to do it numerical.

worksheet

Comment on 2008-07-28 16:22 from zher (8)

How can I send my worksheet to help me in solving problem? (I used maple 11)

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complicated integration

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