unevaluated / lazy series()

[Pablo Puente]

Hi,

I understand that series() supports unevaluated Taylor series setting n=None.

I am writing a Wester test case X3 that probably needs unevaluated Taylor series; series(sin(x))/series(cos(x))==series(tan(x)) . In Mathematica syntax:

In[10]:= (* Note: sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + O(x^9)

         cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + O(x^8)

         tan(x) = x + x^3/3 + 2/15 x^5 + 17/315 x^7 + O(x^9) *)

0. Second

 

In[11]:= tsin = Series[Sin[x], {x, 0, 7}]

0. Second

 

              3    5      7

             x    x      x         8

Out[11]= x - -- + --- - ---- + O[x]

             6    120   5040

 

In[12]:= tcos = Series[Cos[x], {x, 0, 7}]

0. Second

 

              2    4    6

             x    x    x         8

Out[12]= 1 - -- + -- - --- + O[x]

             2    24   720

In[13]:= (* Note that additional terms will be computed as needed *)

0. Second

 

In[14]:= tsin/tcos

0.01 Second

 

              3      5       7

             x    2 x    17 x        8

Out[14]= x + -- + ---- + ----- + O[x]

             3     15     315

 

In[15]:= Series[Tan[x], {x, 0, 7}]

0.01 Second

 

              3      5       7

             x    2 x    17 x        8

Out[15]= x + -- + ---- + ----- + O[x]

             3     15     315

if I try with Sympy  series(sin(x), n=None)/series(cos(x), n=None)) I get TypeError: unsupported operand type(s) for /: 'generator' and 'generator'.

Is i use  series(sin(x))/series(cos(x)) it returns (x - x**3/6 + x**5/120 + O(x**6))/(1 - x**2/2 + x**4/24 + O(x**6))

Is the unevaluated Talyor series support incomplete or is there any other way SymPy can implement this test case?

Thanks,

Pablo Puente

[Aaron Meurer]

My guess is that the paper is looking for the full series, as a
summation. If that's the case, then it isn't implemented.

If it really is looking for an expansion up to some number of terms,
just take (sin(x).series()/cos(x).series()).series() ==
tan(x).series(). Note that unlike Mathematica, SymPy does not
automatically re-compute the series evaluation when you combine series
expansions with other expressions---you need to call series() again.

Aaron Meurer

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[Pablo Puente]

(sin(x).series()/cos(x).series()).series() == tan(x).series() 

worked fine. The tests are combining series expansions.

Thanks Aaron,

Pablo

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[Sergey Kirpichev]

On Monday, October 21, 2013 11:21:40 PM UTC+4, Pablo Puente wrote:

Is the unevaluated Talyor series support incomplete

In general, yes (through, it's easy to workaround as Aaron suggested above).

Actually, SymPy lacks several abstractions to implement things correctly.  We have

very good Gruntz's limit algorithm, but very naive and limited series support for this one.

The series method should return a specific object (not just Add instance), that supports some

appropriate interface (iterator at least).   And the usual arithmetic operations of course.

Some specific cases covers by the formal power/Laurent series with nice algebraic structure (ring). 

PS: Something is in our wiki:

https://github.com/sympy/sympy/wiki/UD-series