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MathGroup Archive 1992

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Re: Integrals involving the Heaviside function

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Re: Integrals involving the Heaviside function
  • From: Roque Donizete de Oliveira <oliveria at engin.umich.edu>
  • Date: Mon, 5 Oct 92 23:56:15 -0400

   Thanks to all of you who responded to my inquire about integrals involving the Heaviside
unit step function (which, by the way, is called UnitStep in MMa 2.1 in <<Calculus`DiracDelta`).
This is one the integrals (in its full and original form) I'm trying to solve symbolically:

Integrate[2 Pi Exp[-a x^2]  UnitStep[y^2 - (x^2 - V^2)/RM]  (y^2 - (x^2 - V^2)/RM)^l  y  Exp[-b y^2] ,
          {x,-Infinity,Infinity},{y,0,Infinity}]

where 
      a, b, RM are real constants > 0;
      V is a real constant >= 0.
      l is an integer constant >=0.

Notice that I don't know the numerical values of a,b,RM,V,l. I seek a symbolic solution.

I have manually solved the integral above (which by the way, is the simplest in its class,
the other being higher order momentum (in x or y) of the integral above). My hand-derived
result is in terms of a sum of Erf (or Hypergeometric1F1) functions. I'm trying to
solve it symbolically for the following reasons:
        1) double checking
        2) perhaps MMa would be able to arrive at a simpler result (without the summation)
        3) to speed up computation of similar integrals once I'm confident I've found a
           CAS that can handle it completely and correctly.

For those who want to start with a simpler integral, for the case V = 0, the result for
the integral above is

                                  l! Pi^(3/2)
                       ------------------------------------
                        b^(l+1) Sqrt[a] Sqrt[1 + b/(a RM)]

Plasma physicists will recognize the integral above as the normalization constant of
a generalized loss cone bi-Maxwellian distribution function, with x (y) being the parallel
(perpendicular) velocity.

I've being able to solve, with MMa 2.1, parts of the integral above (after making some changes
of variables) or particular cases (like setting l=0 and/or V=0).

If someone has a suggestion (even using other CAS) on how to manipulate the integral 
above (in a way that MMa likes) please let me know. 

To see where MMa is failing you should solve one part of the double integral first. 
I've replaced the Heaviside function by the appropriate limits of integration (I didn't
like UnitStep too much) and tried doing the y-integration first and the x-integration first. 
Using the latter, I arrived to an intermediate result (after making a change of variable) like

A  Integrate[ t^(2 l +2) Exp[-B t^2] Hypergeometric1F1[1/2, 3/2 + l,- t^2] , {t,C,Infinity}]
 
where A, B, C are positive real constants. That is where I'm stuck. I don't know if
this last integral is doable or not. If anyone knows of any tricks to solve the integral
above please let me know. 

I've also used the Declare.m to declare the types of a,b,RM,V,l. It seems to help
the integration a little bit.

I realize this is probably not a simple task (although I have to confess I'm neither
mathematician or MMa expert).

Thanks.
  
     Roque
     oliveria at caen.engin.umich.edu
     Nuclear Eng, The University of Michigan





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