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Re: RE: complex analysis problem in mathematica 3.0
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48493] Re: [mg48476] RE: [mg48459] complex analysis problem in mathematica 3.0
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Wed, 2 Jun 2004 04:21:58 -0400 (EDT)
*References*: <NDBBJGNHKLMPLILOIPPOCEBMEBAA.djmp@earthlink.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Of course "by quicker" I meant : requiring less typing ;-)
I do not doubt that MapLevelParts can be very useful, although as a
general principle I prefer to consider each such problem individually
and try to find the most suitable approach rather than rely on a
universal function. My main reason is that doing so is a good practice
in Mathematica programming.
Andrzej
On 1 Jun 2004, at 23:03, David Park wrote:
> I suspected I would get some alternative methods to simplify, but I
> don't
> think that they are all that obvious, and therefore not quicker. I
> still
> think a MapLevelParts command would be useful. It complements MapAt and
> gives much better direct control at manipulating expressions.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
> From: Andrzej Kozlowski [mailto:akoz at mimuw.edu.pl]
To: mathgroup at smc.vnet.net
>
> A quicker way to get the same answer is:
>
> step1 = 1/(s^3 + 2*s^2 + 2*s + 1) /. s -> I*w
>
> Collect[ComplexExpand[step1] /. I -> i, i,
> Simplify] /. i -> I
>
>
> (1 - 2*w^2)/(w^6 + 1) + (I*w*(w^2 - 2))/(w^6 + 1)
>
> P.S. Although it is true one can't use j, one can use the character
> entered as escape j j escape, which looks like a fancy j.
>
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>
>
>
>
> On 1 Jun 2004, at 16:02, David Park wrote:
>
>> You can't use j. Use the Mathematica I.
>>
>> step1 = 1/(s^3 + 2*s^2 + 2*s + 1) /. s -> I*w
>>
>> Then you can separate into real and imaginary parts by using
>> ComplexExpand.
>>
>> step2 = ComplexExpand[step1]
>> 1/((1 - 2*w^2)^2 + (2*w - w^3)^2) -
>> (2*w^2)/((1 - 2*w^2)^2 + (2*w - w^3)^2) +
>> I*(-((2*w)/((1 - 2*w^2)^2 + (2*w - w^3)^2)) +
>> w^3/((1 - 2*w^2)^2 + (2*w - w^3)^2))
>>
>> But how do we further simplify that? We can't use Simplify or Together
>> on
>> the entire expression because that mixes the real and imaginary parts
>> together again. We can get a partial simplification by mapping
>> Simplify onto
>> the parts.
>>
>> step3 = Simplify /@ step2
>> 1/(1 + w^6) - (2*w^2)/(1 + w^6) + (I*w*(-2 + w^2))/
>> (1 + w^6)
>>
>> But the expression could be further simplified by using Together on
>> the
>> first two terms. Unfortunately Mathematica doesn't provide any direct
>> method
>> of doing that other than resorting to a rule that duplicates the parts
>> on
>> the lhs. I think that Mathematica needs an additional routine that
>> would
>> complement MapAt. I call it MapLevelParts and it maps a function to a
>> subset
>> of level parts in an expression. The common application would be to a
>> subset
>> of terms in a Plus expression or a subset of factors in a Times
>> expression.
>>
>> MapLevelParts::usage =
>> "MapLevelParts[function, {topposition, levelpositions}][expr] will
>> map \
>> the function onto the selected level positions in an expression. \
>> Levelpositions is a list of the selected parts. The function is
>> applied to \
>> them as a group and they are replaced with a single new expression.
>> Other \
>> parts not specified in levelpositions are left unchanged.\nExample:\na
>> + b +
>> \
>> c + d + e // MapLevelParts[f, {{2,4,5}}] -> a + c + f[b + d + e]";
>>
>> MapLevelParts[func_,
>> part : {toppart___Integer?Positive,
>> subp : {_Integer?Positive,
>> eprest__Integer?Positive}}][expr_] :=
>> Module[{work, subparts, npos, null, i, nnull = Length[{eprest}]},
>> work = func@Part[expr, Sequence @@ part];
>> subparts = Thread[{toppart, subp}];
>> newparts = {work, Table[null[i], {i, 1, nnull}]} // Flatten;
>> npos = Partition[Range[nnull + 1], 1];
>> ReplacePart[expr, newparts, subparts, npos] /. null[_] ->
>> Sequence[]
>> ]
>>
>> Now we can use Together on just the first two terms of the sum.
>>
>> step3 // MapLevelParts[Together, {{1, 2}}]
>> (1 - 2*w^2)/(1 + w^6) + (I*w*(-2 + w^2))/(1 + w^6)
>>
>> David Park
>>
>
>
>
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