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Topic: Rearranging terms - user-defined
Replies: 5   Last Post: Jul 4, 2012 3:28 AM

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Andrzej Kozlowski

Posts: 2,112
Registered: 1/29/05
Re: Rearranging terms - user-defined
Posted: Jul 4, 2012 3:28 AM
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Well, in this particular case the "normal expression" is just the
polynomial whose terms coincide with the terms of the power series in
degrees less than the degree of the O -term. Whether calling this the
"normal" expression is a good choice of words is not quite clear to
me. The "normal" use of "Normal" is to denote some canonical form of
an expression to which special forms of expressions (of the same kind)
can be reduced but which itself can't be reduced any more. So, for
example, in the case of a graphic object, Normal will express any
GraphicsComplex in terms of the usual graphics primitives. In other
words, a graphic object expressed in a special form is replaced by an
equivalent graphic object expressed in a standard form. Similarly, a
matrix can be expressed in a special form of a SparseArray. There is
no point doing so for an ordinary matrix but if the matrix is
"sparse", working with SparseArray form may lead to a considerable
gain in efficiency. Again, Normal will replace a matrix in SparseArray
form as a standard matrix. But what about Series? Most people seem to
think that Mathematica's series represents, well, a series, but then
how come the "normal" form of a series is a polynomial? There is
something fundamentally illogical here unless we decide that
Series[f,{x,x0,n}} does not represent a series at all (contrary to the
documentation) but only the polynomial made up of the first n terms of
the series. The O part serves only to remind one that the actual
series has (possibly) more terms but still the output of Series is a
polynomial in a special form and not a different kind of mathematical
object altogether. If you adopt this interpretation (that the result
of Series is a polynomial) then there is nothing strange in
Normal[Series….] returning "he same" polynomial in "normal form".

Andrzej Kozlowski

On 2 Jul 2012, at 10:53, Alexei Boulbitch wrote:

Well, in my opinion the best way is:

a + b x - b y + O[x, y]^2 // Normal

a+b (x-y)

Dear Andrzej,

That is a nice method. Could you please kindly comment, what stays
behind. From the documentation it is not quite clear to me. It only
“Normal[expr] converts a power series to a normal expression by
truncating higher-order terms.”
but does not state, what expression is considered to be normal.

Best, Alexei

Alexei BOULBITCH, Dr., habil.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG

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