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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90402] Re: [mg90354] Extracting terms of a polynomial into a list and then
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Tue, 8 Jul 2008 02:28:08 -0400 (EDT)
  • Reply-to: hanlonr at cox.net

expr = 1/(1 - t^2)^(1/2);

int = Integrate[expr, {t, 0, x}, GenerateConditions -> False]

ArcSin[x]

m = 10;

s = Normal[Series[int, {x, 0, m}]];

List @@ s

{x, x^3/6, (3*x^5)/40, (5*x^7)/112, 
   (35*x^9)/1152}

CoefficientList[s, x]

{0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 
   0, 35/1152}

sc = SeriesCoefficient[int, {x, 0, n}]

(Gamma[n/2]*KroneckerDelta[
        Mod[-1 + n, 2]]*UnitStep[
        n - 1])/(n*Sqrt[Pi]*
      Gamma[(n + 1)/2])

Table[sc, {n, 1, m}]

{1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 
   35/1152, 0}


Bob Hanlon

---- Bob F <deepyogurt at gmail.com> 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 + . . .

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 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.

Thanks very much ...

-Bob





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