[Date Index]
[Thread Index]
[Author Index]
Re: asumming and Exp orthogonality condition
*To*: mathgroup at smc.vnet.net
*Subject*: [mg91952] Re: asumming and Exp orthogonality condition
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Mon, 15 Sep 2008 03:39:02 -0400 (EDT)
*Organization*: The Open University, Milton Keynes, UK
*References*: <gag2p0$3ck$1@smc.vnet.net>
charllsnotieneningunputocorreo 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
Prev by Date:
**Re: ListContourPlot3D, no output**
Next by Date:
**Re: Functional programming?**
Previous by thread:
**asumming and Exp orthogonality condition**
Next by thread:
**Re: asumming and Exp orthogonality condition**
| |