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## Re: [Axiom-math] repeated series expansion

 From: Ralf Hemmecke Subject: Re: [Axiom-math] repeated series expansion Date: Fri, 16 Jul 2010 12:22:14 +0200 User-agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.9.2.4) Gecko/20100608 Thunderbird/3.1

I guess, before somebody can give you any answer, you would have to specify what expression do you want as an answer.

series(  series(  sin(x+y),x=0  ),y=0   )

This doesn't work, but it looks as if you want to expand sin(x+y) into a series in x with some "strange" coefficients.

series(  sin(x+y),x=0)

(1)
sin(y)  2   cos(y)  3   sin(y)  4   cos(y)  5
sin(y) + cos(y)x - ------ x  - ------ x  + ------ x  + ------ x
2           6          24          120
+
sin(y)  6   cos(y)  7   sin(y)  8   cos(y)  9    sin(y)  10      11
- ------ x - ------ x + ------ x + ------ x - ------- x + O(x ) 720 5040 40320 362880 3628800
Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)

and then want to expand each coefficient into a series in y.

The result would then be a series in x whose coefficients are *series* in y. Is that what you want?

Then you would go this way...

TY := UnivariateTaylorSeries(Fraction Integer,'y,0)
TX := UnivariateTaylorSeries(TY,'x,0)
sx := taylor(sin(x+y),x=0)
coeffs := [taylor(t,y=0)::TY for t in coefficients sx]
s := series(coeffs)\$TX

That looks a bit complicated, but can you have that in another CAS? Series with coefficients being series themselves? And these are no finite objects. You can ask the resulting series for any coefficient you like.

But I guess, you wanted something else.

Ralf

(1) -> TY := UnivariateTaylorSeries(Fraction Integer,'y,0)

(1)  UnivariateTaylorSeries(Fraction(Integer),y,0)

Type: Type
(2) -> (2) -> TX := UnivariateTaylorSeries(TY,'x,0)

(2)

UnivariateTaylorSeries(UnivariateTaylorSeries(Fraction(Integer),y,0),x,0)

Type: Type
(3) -> sx := taylor(sin(x+y),x=0)

(3)
sin(y)  2   cos(y)  3   sin(y)  4   cos(y)  5
sin(y) + cos(y)x - ------ x  - ------ x  + ------ x  + ------ x
2           6          24          120
+
sin(y)  6   cos(y)  7   sin(y)  8   cos(y)  9    sin(y)  10      11
- ------ x  - ------ x  + ------ x  + ------ x  - ------- x   + O(x  )
720        5040        40320      362880      3628800
Type: UnivariateTaylorSeries(Expression(Integer),x,0)
(4) -> coeffs := [taylor(t,y=0)::TY for t in coefficients sx]

(4)
1  3    1   5     1   7      1    9      11
[y - - y  + --- y  - ---- y  + ------ y  + O(y  ),
6      120      5040      362880
1  2    1  4    1   6     1    8      1     10      11
1 - - y  + -- y  - --- y  + ----- y  - ------- y   + O(y  ),
2      24      720      40320      3628800
1      1  3    1   5     1    7      1    9      11
- - y + -- y  - --- y  + ----- y  - ------ y  + O(y  ),
2     12      240      10080      725760
1    1  2    1   4     1   6      1    8       1     10      11
- - + -- y  - --- y  + ---- y  - ------ y  + -------- y   + O(y  ),
6   12      144      4320      241920      21772800
1      1   3     1   5      1    7      1     9      11
-- y - --- y  + ---- y  - ------ y  + ------- y  + O(y  ),
24     144      2880      120960      8709120
1     1   2     1   4     1    6      1     8       1      10      11
--- - --- y + ---- y - ----- y + ------- y - --------- y + O(y ),
120   240      2880      86400      4838400      435456000
1        1   3     1    5      1     7       1      9      11
- --- y + ---- y  - ----- y  + ------- y  - --------- y  + O(y  ),
720     4320      86400      3628800      261273600

1      1    2      1    4      1     6       1      8
- ---- + ----- y  - ------ y  + ------- y  - --------- y
5040   10080      120960      3628800      203212800
+
1       10      11
----------- y   + O(y  )
18289152000
,
1 1 3 1 5 1 7 1 9 11 ----- y - ------ y + ------- y - --------- y + ----------- y + O(y ),
40320     241920      4838400      203212800      14631321600

1        1    2      1     4       1      6        1       8
------ - ------ y  + ------- y  - --------- y  + ----------- y
362880   725760      8709120      261273600      14631321600
+
1        10      11
- ------------- y   + O(y  )
1316818944000
,
...]
Type: Stream(UnivariateTaylorSeries(Fraction(Integer),y,0))
(5) -> series(coeffs)\$TX

(5)
1  3    1   5     1   7      1    9      11
y - - y  + --- y  - ---- y  + ------ y  + O(y  )
6      120      5040      362880
+
1  2    1  4    1   6     1    8      1     10      11
(1 - - y  + -- y  - --- y  + ----- y  - ------- y   + O(y  ))x
2      24      720      40320      3628800
+
1      1  3    1   5     1    7      1    9      11   2
(- - y + -- y  - --- y  + ----- y  - ------ y  + O(y  ))x
2     12      240      10080      725760
+
1    1  2    1   4     1   6      1    8       1     10      11   3
(- - + -- y  - --- y  + ---- y  - ------ y  + -------- y   + O(y  ))x
6   12      144      4320      241920      21772800
+
1      1   3     1   5      1    7      1     9      11   4
(-- y - --- y  + ---- y  - ------ y  + ------- y  + O(y  ))x
24     144      2880      120960      8709120
+
1 1 2 1 4 1 6 1 8 1 10 11 5 (--- - --- y + ---- y - ----- y + ------- y - --------- y + O(y ))x
120   240      2880      86400      4838400      435456000
+
1        1   3     1    5      1     7       1      9      11   6
(- --- y + ---- y  - ----- y  + ------- y  - --------- y  + O(y  ))x
720     4320      86400      3628800      261273600
+
1      1    2      1    4      1     6       1      8
- ---- + ----- y  - ------ y  + ------- y  - --------- y
5040   10080      120960      3628800      203212800
+
1       10      11
----------- y   + O(y  )
18289152000
*
7
x
+
1          1    3      1     5       1      7        1       9
----- y - ------ y  + ------- y  - --------- y  + ----------- y
40320     241920      4838400      203212800      14631321600
+
11
O(y  )
*
8
x
+
1        1    2      1     4       1      6        1       8
------ - ------ y  + ------- y  - --------- y  + ----------- y
362880   725760      8709120      261273600      14631321600
+
1        10      11
- ------------- y   + O(y  )
1316818944000
*
9
x
+
1            1     3       1      5        1       7
- ------- y + -------- y  - --------- y  + ----------- y
3628800     21772800      435456000      18289152000
+
1        9      11
- ------------- y  + O(y  )
1316818944000
*
10
x
+
11
O(x  )
Type: UnivariateTaylorSeries(UnivariateTaylorSeries(Fraction(Integer),y,0),x,0)