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Re: Extracting terms of a polynomial into a list and then multiply

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90381] Re: Extracting terms of a polynomial into a list and then multiply
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 8 Jul 2008 02:24:10 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g4sm99$p$1@smc.vnet.net>

Bob F wrote:

> Can anyone suggest a way to extract the terms of a polynomial into a
> list. For example the integral of the series expansion of
> 
>              1
>     --------------------
>     (1 - t^2) ^(1/2)
> 
> could be expressed in Mathematica (the first 50 terms) as
> 
>       Integrate[Normal[Series[(1 - t^2)^(-1/2), {t, 0, 50}]], {t, 0,
> x}]
> 
> and gives the polynomial
> 
>     x + x^3/6 + (3 x^5)/40 + (5 x^7)/112 + (35 x^9)/1152 + (63 x^11)/
> 2816 + (231 x^13)/13312 + (143 x^15)/10240 +
>          (6435 x^17)/557056 + (12155 x^19)/1245184 + (46189 x^21)/
> 5505024 + . . .

Let's call the above expression "poly".

> And I would like to extract each term of this polynomial into a list
> like
> 
>     { x, x^3/6, (3 x^5)/40, (5 x^7)/112, (35 x^9)/1152, (63 x^11)/
> 2816, (231 x^13)/13312, (143 x^15)/10240,
>          (6435 x^17)/557056,  (12155 x^19)/1245184,  (46189 x^21)/
> 5505024,  . . . }

Then, we change the *head* of the expression poly to a List thanks to 
the command Apply:

     List @@ poly

> Then I would like to take this list and multiply each element in the
> list by the integrated polynomial in order to get a list of
> polynomials that shows all of the components of the fully multiplied
> polynomial in an expanded form. In other words I would like to show
> the term by term expansion of the integral multiplied by itself, ie
> 
>      Expand[ Integrate[Normal[Series[(1 - t^2)^(-1/2), {t, 0, 50}]],
> {t, 0, x}] *
>                   Integrate[Normal[Series[(1 - t^2)^(-1/2), {t, 0,
> 50}]], {t, 0, x}]]
> 
> Was working thru an example of what Euler did to compute Zeta[2] and
> was looking for patterns in the polynomial coefficients.

The result you are looking for should be

     Expand[terms*poly]

since lists (terms) are automatically *threaded* over a single 
expression (poly) when multiplied (Times), and Expand is *listable*, 
i.e. it is automatically applied to each element of a list.

The resulting one-liner is

     (Expand[List @@ #1*#1] & )[Integrate[
          Normal[Series[(1 - t^2)^(-2^(-1)), {t, 0, 50}]], {t, 0, x}]]


Here is a detailed example (with the first 10 terms of the series 
expansion).

In[1]:= poly = Integrate[Normal[Series[(1 - t^2)^(-2^(-1)), {t, 0, 
10}]], {t, 0, x}]

Out[1]=
      3      5      7       9       11
     x    3 x    5 x    35 x    63 x
x + -- + ---- + ---- + ----- + ------
     6     40    112    1152     2816

In[2]:= terms = List @@ poly

Out[2]=
      3     5     7      9      11
     x   3 x   5 x   35 x   63 x
{x, --, ----, ----, -----, ------}
     6    40   112   1152    2816

In[3]:= Expand[terms*poly]

Out[3]=
        4      6      8       10       12
   2   x    3 x    5 x    35 x     63 x
{x  + -- + ---- + ---- + ------ + ------,
       6     40    112     1152     2816

    4    6    8      10       12       14
   x    x    x    5 x     35 x     21 x
   -- + -- + -- + ----- + ------ + ------,
   6    36   80    672     6912     5632

      6    8      10      12      14        16
   3 x    x    9 x     3 x     7 x     189 x
   ---- + -- + ----- + ----- + ----- + -------,
    40    80   1600     896    3072    112640

      8      10      12       14       16       18
   5 x    5 x     3 x     25 x     25 x     45 x
   ---- + ----- + ----- + ------ + ------ + ------,
   112     672     896    12544    18432    45056

       10       12      14       16         18        20
   35 x     35 x     7 x     25 x     1225 x     245 x
   ------ + ------ + ----- + ------ + -------- + -------,
    1152     6912    3072    18432    1327104    360448

       12       14        16       18        20         22
   63 x     21 x     189 x     45 x     245 x     3969 x
   ------ + ------ + ------- + ------ + ------- + --------}
    2816     5632    112640    45056    360448    7929856


Regards,
-- Jean-Marc


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