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Re: D under Sum

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  • Subject: [mg128400] Re: D under Sum
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Sat, 13 Oct 2012 01:02:59 -0400 (EDT)
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Am 11.10.2012 08:23, schrieb Nicolas Venuti:
> Hello
>
> I want the derivation D to be applied under the Sum sign,
> I have a function HoldSum that do not evaluate but is similar to Sum in form
>
> In[3]:= D[HoldSum[p*i, {i, 1, n}], p]
>
> I get the output :
>
> Out[3]=i*HoldSum^(1, {0, 0, 0})[i p, {i, 1, n}]
>
> but I would want that D be applied under the Sum sign like done by hand here :
>
> In[5]:= HoldSum[D[p i, i], {i, 1, n}]
>
> Out[5]= HoldSum[p, {i, 1, n}]
>
> Obviously I want D to work for any expression containing HoldSum (even on nested HoldSum expression)
>
> Does anybody knows how to do that ?

As you have observed already, manipulating expressions in sums and 
integrals with respect to the summation variable seems to be 
mathematical nonsense, because the name of the domain construct is local 
and the sums are constants with respect to them.

A path to implemt your goal is, as in functional and fourier analysis as 
well as quantum maechanics, to switch to a general context of a domain 
space with a fixed variable name and apply transformation rules to the 
abstract sum term together with the domain list.

The  container "List" suffices to implement the two main operations, 
that make up the Heisenberg algebra on the space of eg

{summand,{i,0,inf}}

constructs:

Applying a function f of two variables f: {i,term} ->f[i,term]

apply[f_][{summand_,domain_List}] := {f[domain[[1]],summand],domain}

and derivaitve with respect to the summation variable

derivative[{summand_,domain_}]:={D[summand,domain[[1]]],{domain}}

Evaluation of sums and integrals is considered as a linear operation 
sending the List to the domain of values ot the terms

eval{[term_,domain_List}]:= Sum[term,domain]

Logically, after symbolic evalutation the result is considered as a 
number. Numbers cannot be changed by manipulation of the summation term, 
even if evaluation result does not yield a number. With the exception of 
identical and simplifying transformations.

The last case is of course a theme in the Mathematica simplification 
process because resummation technique requires complex replacements of 
summation term and the domain construct in unevaluated sums and 
integrals with fixed summation domains.

-- 

Roland Franzius



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