RE: complex analysis problem in mathematica 3.0

• To: mathgroup at smc.vnet.net
• Subject: [mg48476] RE: [mg48459] complex analysis problem in mathematica 3.0
• From: "David Park" <djmp at earthlink.net>
• Date: Tue, 1 Jun 2004 03:02:51 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```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];
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

From: BranasMan [mailto:branasREmoVe at mail.inet.hr]
To: mathgroup at smc.vnet.net

i have a complex function:

H(s)=1 / (s^3 + 2s^2 + 2s + 1)

whan i replace "s" with j*w (j=sqrt(-1)) i get:

H=1 / (1 + j2w -2w^2 - jw^3)

i would like to get that function in shape of :
H=something + j*something_else i.e. the complex
and real part apart.

i played with Re and Im,but it seems that the fact that
"w" is a variable confuses mathematica?!

i would reeeealy appreciate any help,and maybe perhaps some
links for me to learn to use mathematica better.

```

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