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MathGroup Archive 1995

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Re: Few MMA questions - need some help!

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1074] Re: Few MMA questions - need some help!
  • From: rubin at msu.edu (Paul A. Rubin)
  • Date: Fri, 12 May 1995 17:41:02 -0400
  • Organization: Michigan State University

In article <3o6u1g$psn at news0.cybernetics.net>,
   pikus at dolphin.physics.ucsb.edu (Fedor G. Pikus) wrote:
->Hi,
->I need some help to make MMA do the calculation I want to:
->
->1. MMA 2.2 cannot take the following integral:
->	Integrate[E^(s t) E^(I wmk t),{t,-Infinity,T}]
->even if I write 
->	Integrate[E^(s^2 t) E^(I wmk t),{t,-Infinity,T}]
->which defines that at -Infinity Exp goes to 0, it still cannot take it.
->MMA 2.0 did take this integral, and the one before, where it assumed
->positive s. 

This may be a bug in Integrate.  I verified that kernel 2.2.2 cannot do it 
either; but if you combine the factors in the integrand, you get an answer:

  In[1]:=  Integrate[ E^((s + I wkm) t), {t, -Infinity, T} ]
  Out[1]=
       T (s + I wkm)
      E
      --------------
         s + I wkm

->The integral with finite limits 
->	Integrate[E^(s^2 t) E^(I wmk t),{t,-A,T}]
->can be computed, but the limit cannot. What it so difficult about 
->limit of E^(s^2 t) at t-> -Infinity?

Actually, I was about to say that I'm not sure the answer above is 
appropriate.  You may be thinking that s is real, but Mathematica doesn't 
know that (and perhaps should not assume it).  I assume you feel the limit 
is 0; but what if s = I?

->2. I need to make MMA compute the integrals like this:
->	Int[t Sum[f[k],{k,Infinity}],{t,0,1}]
->and
->	Int[If[m == n, t, t^2], {t,0,1}]
->to 
->	Sum[f[k],{k,Infinity}]
->and
->	If[m == n, 1, 1/2]
->respectively.

You can write your own integration function to do all this:

In[1]:=  Clear[ int ];
In[2]:=  int[ x_ Sum[ z__ ], y_List ] := 
           Sum[ z ] Integrate [x, y ] /; FreeQ[ z, y[[1]] ]
In[3]:=  int[ If[ a_, b_, c_ ], d_List ] :=
           If[ a, 
             Evaluate[ Integrate[ b, d ] ], 
             Evaluate[ Integrate[ c, d ] ] 
           ]
In[4]:=  int[ x_, y_List ] := Integrate[ x, y ]

In[5]:=  int[ t Sum[ f[k], {k, Infinity} ], {t, 0, 1} ]
Out[5]=  
         Sum[f[k], {k, Infinity}]
         ------------------------
                    2
In[6]:=  int[ If[ m== n, t, t^2 ], {t, 0, 1} ]
Out[6]=
                     1  1
          If[m == n, -, -]
                     2  3

Paul

**************************************************************************
* Paul A. Rubin                                  Phone: (517) 432-3509   *
* Department of Management                       Fax:   (517) 432-1111   *
* Eli Broad Graduate School of Management        Net:   RUBIN at MSU.EDU    *
* Michigan State University                                              *
* East Lansing, MI  48824-1122  (USA)                                    *
**************************************************************************
Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE


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