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MathGroup Archive 2007

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re: Same Limit in v5.2 and v6.0 - conclusion, I hope...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78004] re: Same Limit in v5.2 and v6.0 - conclusion, I hope...
  • From: jrc <jrchaff at mcn.net>
  • Date: Thu, 21 Jun 2007 05:42:54 -0400 (EDT)

This may be of use to others, so I furnish the conclusion,
prompted by yet another offline communication. Here is the
code that works in v6.0: (I have been advised to post 'actual
mathematica code' - so some lines may have to be text...)

In[1]= $Assumptions = a > 0 && k1 > 0 && k2 > 0 && Element[x, Reals]

Out[1]= a > 0 && k1 > 0 && k2 > 0 && x e(in) Reals

In[2]= i1b = (1/a)*Integrate[Exp[-i*k2*x]*Exp[i*k1*x],x]

               2*Sin[(1/2)*a*(k1-k2)]
Out[2]=      ------------------------
                      a*(k1-k2)

In[3]=  Limit[i1b, a -> Infinity]

Out[3]= 0
----------------
This is the obviously correct result, since it is the same
limit as sin(x)/x at infinity.

However, the use of $Assumptions is in fact different in v6.0,
and I can find no instance in the 'new' 6.0 documentation that
shows this, including the little 'new in 6' links.

In v5.2, the assumptions statement, shown in a number of
books,

$Assumptions = {a > 0, k1 > 0, k2 > 0, x e(in) Reals}

works perfectly fine. In fact, the use of a list enclosed in
brackets is shown in one of the examples in the 5.2 Help Browser
rather clearly under '$Assumptions'. But this form will not
work in 6.0 until three modifications are made:

1) The list should be joined with Boolean 'and': "&&" operators;

2) Apparently one must use the 'Element' function instead of the
   funny 'e' symbol (not sure this is necessary);

But finally, strangely enough and this *IS* a change from 5.2,
enclosing the list in brackets does not work in 6.0 - at least for
the 'Limit' function.

Once again, I would like to see a clear statement of differences
in 6.0 for the oft-used functions. "$Assumptions" falls into this
category.

jrc


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