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Re: asumming and Exp orthogonality condition

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91990] Re: asumming and Exp orthogonality condition
  • From: charllsnotieneningunputocorreo at gmail.com
  • Date: Tue, 16 Sep 2008 19:23:28 -0400 (EDT)
  • References: <gag2p0$3ck$1@smc.vnet.net> <gal3eg$drq$1@smc.vnet.net>

On Sep 15, 2:38 am, Jean-Marc Gulliet <jeanmarc.gull... at gmail.com>
wrote:
> charllsnotieneningunputocor... at gmail.com wrote:
> > Element[ 0 , Integers ] evaluates to true, however;
>
> > Assuming[ Element[ n , Integers] , Integrate[ Exp [ 2 I Pi n x ] ,
> > {x , 0 , 1 } ] ] evaluates to zero. Shouldn't evaluate to
> > KroneckerDelta[ 0 , n ] instead?
>
> The following might explain what is going on. Mathematica does not have
> any transformation rule for this specific case of this definite integral.
>
> So, I guess, what Mathematica computes first is the general formula for
> the definite integral (I shall call it int[n]) and then applies the
> assumption about n being an integer. (This is conceptually equivalent to
> In[1] and In[2].)
>
> As it stands, int[n] is defined and equal to 0 for all n in N, n !=0.
> (For n == 0 we have a division by zero.) So what Mathematica sees is
> that the function is defined for every non-zero integer and its value is
> therefore zero.
>
> Now, if we extend the domain of definition of int[n] to the whole set of
>   integers and defined int[0] == 1, (having checked that the limit =
of
> int[n] as n approaches zero on the left and on the right is equal to
> one), only then this extended definition matches KroneckerDelta[0, n].
>
> Thus, Mathematica's behavior seems reasonable since Mathematica is not
> going to attempt by itself to check the limits and/or extend the domain
> of definition.
>
> In[1]:= int[n_] = Integrate[Exp[2 I Pi n x], {x, 0, 1}]
>
> Out[1]= -((I (-1 + E^(2 I n \[Pi])))/(2 n \[Pi]))
>
> In[2]:= Assuming[Element[n, Integers], Simplify[int[n]]]
>
> Out[2]= 0
>
> In[3]:= Table[int[n], {n, -2, 2}]
>
> During evaluation of In[3]:= Power::infy: Infinite expression 1/0 \
> encountered. >>
>
> During evaluation of In[3]:= \[Infinity]::indet: Indeterminate \
> expression (0 ComplexInfinity)/\[Pi] encountered. >>
>
> Out[3]= {0, 0, Indeterminate, 0, 0}
>
> In[4]:= Limit[int[n], n -> 0, Direction -> 1]
>
> Out[4]= 1
>
> In[5]:= Limit[int[n], n -> 0, Direction -> -1]
>
> Out[5]= 1
>
> Regards,
> -- Jean-Marc


Interesting, thanks for your reply

I've read that it is possible to unprotect the Integrate function and
add/override values with custom functions. Is it possible to override
this in this particular case?


thanks,
Charles J. Quarra


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