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MathGroup Archive 2000

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Re: Sums and Products: Compact Notation and Differentiation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23941] Re: Sums and Products: Compact Notation and Differentiation
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Fri, 16 Jun 2000 00:57:27 -0400 (EDT)
  • Organization: Universitaet Leipzig
  • References: <8i9qoc$2oh@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

it is possible but you have to do it by your self because 
simply

Clear[sum]
sum /: D[sum[a_, range_], x_] /; FreeQ[range, x] := sum[D[a, x], {i, n}]

will do it but

a) what is with the index shifts ? the {i,n} iterator mean {i,1,n} and
   this works only for the first derivative of the sum, with a sum
    over {i,0,n}  your equations is not true and for a power series
    one have
    D[sum[a_*x^i_,{i_,n1_,n2_}],x] /; FreeQ[{i,n1,n2},x]:=
       sum[D[a*x^i],x],{i,n1+1,n2}]
    because the i=0 term depends not on x
b) how to "normalize" expressions like
   sums of sum[] ? collect them ? shift the index in the terms
   until one can make a sum[a,i]+sum[b,i] :> sum[a+b,i]

Regards
  Jens

Justus Piater wrote:
> 
> Hi,
> 
> I am relatively new to Mathematica (Version 3.0), and have been able
> to get sensible results using expressions involving Sum and Product
> symbols only if the summation/multiplication ranges are given by
> constants. I have the following specific problems:
> 
> 1. When I have Mathematica operate on a formula involving Sum and
>    Product symbols with constant ranges, it always displays the result
>    in expanded form, without these symbols, even though it often seems
>    easily possible and much more compact to use Sums and Products. Is
>    is possible to have Mathematica retain the convenient notation, and
>    prevent it from expanding the terms?
> 
> 2. I can't get Mathematica to give me simple derivatives of
>    expressions involving Sums and Products, unless the ranges are
>    known to Mathematica so it can expand them. For example, I would
>    like to type:
> 
>        D[Sum[Subscript[a,i]x^i,{i,n}],x]
> 
>    and get as output some rendering of:
> 
>        Sum[i Subscript[a,i]x^(i-1),{i,n}]
> 
>    Is this really not possible?
> 
> Thanks,
> Justus
> 
> --
> Justus Piater              Laboratory for Perceptual Robotics   U of Mass
> www.cs.umass.edu/~piater       Computer Vision Laboratory         Amherst


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