ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 18 Sep 2013 14:38:36 -0500integrating formal Laurent serieshttp://ask.sagemath.org/question/10546/integrating-formal-laurent-series/I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. It gets pretty messy so I was hoping that sage could keep track of the details for me. Here are the things I don't know how to do:
1) Define a Laurent series by giving an expression for its n-th coefficient.
2) Formally multiply and integrate Laurent series, producing an explicit expression for the n-th coefficient of the result.
Is this possible? I apologize if some or all of this is explained elsewhere.
EDIT: An example of what I want to do would be to define a power series such as sum(n * x^n,n,0,infinity) and then integrate it and/or multiply it by another power series, resulting in yet another power series whose coefficients I can read off. The O(x^n) notation is not very useful to me since I would like to have an expression for an arbitrary coefficient of that resulting series in terms of n.Wed, 18 Sep 2013 12:39:16 -0500http://ask.sagemath.org/question/10546/integrating-formal-laurent-series/Answer by Luca for <p>I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. It gets pretty messy so I was hoping that sage could keep track of the details for me. Here are the things I don't know how to do:</p>
<p>1) Define a Laurent series by giving an expression for its n-th coefficient.</p>
<p>2) Formally multiply and integrate Laurent series, producing an explicit expression for the n-th coefficient of the result.</p>
<p>Is this possible? I apologize if some or all of this is explained elsewhere.</p>
<p>EDIT: An example of what I want to do would be to define a power series such as sum(n * x^n,n,0,infinity) and then integrate it and/or multiply it by another power series, resulting in yet another power series whose coefficients I can read off. The O(x^n) notation is not very useful to me since I would like to have an expression for an arbitrary coefficient of that resulting series in terms of n.</p>
http://ask.sagemath.org/question/10546/integrating-formal-laurent-series/?answer=15471#post-id-15471It is not clear to me what you mean by "giving an expression for the n-th coefficient". Is this what you are looking for?
sage: R.<z> = FractionField(QQ[['z']])
sage: R
Laurent Series Ring in z over Rational Field
sage: (z^-2 + z^3 + 5*z^8 + O(z^10)).integral()
-z^-1 + 1/4*z^4 + 5/9*z^9 + O(z^11)
Wed, 18 Sep 2013 14:38:36 -0500http://ask.sagemath.org/question/10546/integrating-formal-laurent-series/?answer=15471#post-id-15471