Francois Maltey <[EMAIL PROTECTED]> writes:
> Dear Martin,
>
> I'm not proud about series but let me tell you what I use in series.
>
> //1// I only use series in analysis,
> so series with severe variables are very rare for me.
>
> There are some
>
> sum(in x^i) sum(in y^j) ... x^i y^j = sum(in y^j) sum(in x^j) ...
>
> but this use of axiom isn't so fine because it's impossible to find the
> function from this serie in axiom.
Sorry, you need to be more explicit. What do you mean with
> sum(in x^i) sum(in y^j) ... x^i y^j = sum(in y^j) sum(in x^j) ...
and what do you mean with
> to find the function from this serie in axiom.
???
> //4// Sometime I compute the recip of y=1+x+x^3 arround x=0. So I get a serie
> x=1-y+... Do you know if there is this revert function in axiom.
> It's not recip that is equal to 1/...
Do you mean "revert", i.e. the compositional inverse?
(58) -> res := revert(series(x*(1+ x + x^3), x=0))
(58)
2 3 4 5 6 7 8 9 10 11
x - x + 2x - 6x + 20x - 70x + 256x - 969x + 3762x - 14894x + O(x )
Type: UnivariateTaylorSeries(Expression(Integer),x,0)
(59) -> res + res^2+res^4
11
(59) x + O(x )
Type: UnivariateTaylorSeries(Expression(Integer),x,0)
The series expansion of the root of 1+ x + x^3 itself is ugly, and revert
expects that the constant term vanishes...
> //5// I use the coefficient function with polynom.
> By example M(a,0) and N(0,b) are points. Give a vector of (MN) line.
>
> The equation of the line is
> eqline := determinant matrix [[a,0,1],[0,b,1],[x,y,1]] (equal 0)
>
> A vector is vector [coefficient (eqline, y, 1), - coefficient (eqline, x, 1)].
>
> I don't know if this method may be used for series.
*What* do you want to do with series?
>
> // 6 // An other exercice I do whith series is to find a naive way for
> the serie of the nth root of x=tan x when x -> %plusInfinity.
>
> So we write cos x * x = sin x and x = n*%pi + %pi/2 - a/n - b/n^2 ...
> You write the two series and find first a, then b, and so.
>
> ** I don't try this exercice with axiom ** Sometime I do it with maple.
Maybe you could show us how you do this with maple?
>
> //7// Of corse some series I use aren't around 0 or have a not integer
> power...
>
> //8// I quickly try to test the silly series (x^a-a^x) and (x^x - a^a) for a
> fixed a when x is arround a but I can't with axiom.
(67) -> series(x^a-a^x, x=a)
(67)
2 a log(a)
a log(a) (- a log(a) + a - 1)%e 2
(- log(a) + 1)%e (x - a) + ------------------------------- (x - a)
2a
+
2 3 2 a log(a)
(- a log(a) + a - 3a + 2)%e 3 4
------------------------------------- (x - a) + O((x - a) )
2
6a
Type: UnivariatePuiseuxSeries(Expression(Integer),x,a)
(68) -> series(x^x-a^a, x=a)
(68)
a log(a) a a log(a)
%e - a + (log(a) + 1)%e (x - a)
+
2 a log(a)
(a log(a) + 2a log(a) + a + 1)%e 2 3
----------------------------------------- (x - a) + O((x - a) )
2a
Type: UnivariatePuiseuxSeries(Expression(Integer),x,a)
(69) -> series(x^x-exp(a*log a), x=a)
(69)
a log(a)
(log(a) + 1)%e (x - a)
+
2 a log(a)
(a log(a) + 2a log(a) + a + 1)%e 2
----------------------------------------- (x - a)
2a
+
2 3 2 2 2 2 a log(a)
(a log(a) + 3a log(a) + (3a + 3a)log(a) + a + 3a - 1)%e
-------------------------------------------------------------------
2
6a
*
3
(x - a)
+
4
O((x - a) )
Type: UnivariatePuiseuxSeries(Expression(Integer),x,a)
Martin
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