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