Expansion of binomials with Rational coeff

[Sachin Joglekar]

In the special case where, in the expression (x+y)**n , n is non-integer Rational, shouldn't the expansion come out to be a series of the form

y**n+n*x*y**n-1+0.5*n*(n-1)*(x**2)*(y**n-2)+...+O(x**6) ?

Maybe this shouldn't be the default return for expand, but it would be nice to have the option of getting that particular binomial series, especially if value of n is known.

Something like (x+y+z)**n could be written by substituting (y+z) for y in the above expansion  (Thats how WolframAlpha shows it) or we could use recursion to expand each term of the form (y+z)**(n-r)

[Sachin Joglekar]

Just finished coding a method (named it 'expansion') to giv the above desired result. Example of input-output ->

>>> expansion((x+y)**2)

x**2 + 2*x*y + y**2

>>> expansion((3*a+2*b)**2)

9*a**2 + 12*a*b + 4*b**2

>>> expansion((x+y)**0.2)

x**0.2 + 0.2*x**-0.8*y - 0.08*x**-1.8*y**2 + 0.048*x**-2.8*y**3 - 0.0336*x**-3.8*y**4 + 0.025536*x**-4.8*y**5 + O(y**6)

>>> expansion((3*x+y+z+2)**3.4)

(3*x + z + 2)**3.4 + 3.4*y*(3*x + z + 2)**2.4 + 4.08*y**2*(3*x + z + 2)**1.4 + 1.904*y**3*(3*x + z + 2)**0.4 + 0.1904*y**4*(3*x + z + 2)**-0.6 - 0.022848*y**5*(3*x + z + 2)**-1.6 + O(y**6)

[Ondřej Čertík]

I think the only other way is to use series(). Would you mind sending
this as a github pull request?

Ondrej

[Christophe BAL]

Hello,

expansion((x+y)**0.2) should raise one error.

Christophe

[Sachin Joglekar]

I will send the request shortly. Even I used the series function, but it had to be processed to accomodate the presence of more than one variables in the expression.

[Sachin Joglekar]

The only question I have is how should I accomodate this functionality? Should I modify the expand() function or write this as a separate function?

[Sachin Joglekar]

@Christophe BAL ..Why would (x+y)**0.2 raise an error? I guess the binomial expansion of it would be an infinite series

[Aaron Meurer]

I'd put it as a separate function, or at least as a non default hint to expand. And if it goes in expand, it should give the whole summation. If it's just a series (with an O term) it should go in series somewhere. 

Aaron Meurer

On Oct 8, 2012, at 3:07 AM, Sachin Joglekar <srjogl...@gmail.com> wrote:

The only question I have is how should I accomodate this functionality? Should I modify the expand() function or write this as a separate function?

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[Sachin Joglekar]

So we could just add a function say like 'binomial_series' in the series module?

[Sachin Joglekar]

And what is the best way, if any, to express the summation of an infinite series? That would be a nice way to represent the entire expression.

[Aaron Meurer]

Just use Sum(), and whatever combinatorial functions are necessary. We should have all the important ones. For the summation variable, use Dummy so that you know it won't clash with a symbol name used by the user. 

Aaron Meurer

On Oct 8, 2012, at 10:31 AM, Sachin Joglekar <srjogl...@gmail.com> wrote:

And what is the best way, if any, to express the summation of an infinite series? That would be a nice way to represent the entire expression.

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