On Dec 4, 2007, at 5:09 AM, fwc wrote:
>>> 1) Taylor series of a rational function.
>>
>>> This works:
>>> sage: cos(x).taylor(x,0,2)
>>
>>> This doesn't:
>>> sage: x/(1+x).taylor(x,0,2)
>>
>>> This is very confusing:
>
>> This is due to the fact that '.' binds tighter than '/'. For
>> example,
>> sage: x/(1+x).taylor(x,0,2)
>> x/(x + 1)
>> sage: x/((1+x).taylor(x,0,2))
>> x/(x + 1)
>> sage: (x/(1+x)).taylor(x,0,2)
>> x - x^2
>>
>> In Sage, "(x/(1+x))" creates an object and the you call the taylor()
>> method on that object.
>
> Mathematica has the advantage that Series creates a truncated series
> object rather than a polynomial. Thus it doesn't matter whether the
> division is done before or after:
>
> sage: mathematica("x/Series[1+x, {x, 0, 1}]")
> SeriesData[x, 0, {1, -1}, 1, 3, 1]
> sage: mathematica("Series[x/(1+x), {x, 0, 2}]")
> SeriesData[x, 0, {1, -1}, 1, 3, 1]
Hmmmm this is an excellent point. We do have a PowerSeriesRing which
can keep track of where you truncated to, but it's only used in a
more strictly algebraic setting, it's not really part of the symbolic
calculus package. Is it possible for the symbolic calculus package to
do something similar to this? What about creating a PowerSeriesRing
with the SymbolicExpressionRing as the base ring?
sage: R.<z> = PowerSeriesRing(SymbolicExpressionRing)
------------------------------------------------------------------------
---
<type 'exceptions.TypeError'> Traceback (most recent call
last)
/Users/david/<ipython console> in <module>()
/Users/david/sage-2.8.14/local/lib/python2.5/site-packages/sage/rings/
power_series_ring.py in PowerSeriesRing(base_ring, name,
default_prec, names, sparse)
171 R = PowerSeriesRing_generic(base_ring, name,
default_prec, sparse=sparse)
172 else:
--> 173 raise TypeError, "base_ring must be a commutative ring"
174 _cache[key] = weakref.ref(R)
175 return R
<type 'exceptions.TypeError'>: base_ring must be a commutative ring
Well maybe not....
Would be nice though....
david
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