UD - Function expansions and series - roadmap

Note: these are just the ideas of one person. It may not necessarily coincide with the ideas of others in the community.

Note: For GSoC, it doesn't have to be this way. On the contrary, we welcome the participants to describe their own ideas.

The main directions of extension

(which we must take into account) widely are:

1. kinds of exponents:

   • non-zero x0 point (Taylor vs McLaren)
   • negative exponents (Laurent)
   • multivariate
   • generalized exponents (rational, complex exp)

2. kinds of coefficients:

   • rings ZZ, QQ etc

3. kinds of bases and polynomials bases:

   • Power
   • trigonometric series
   • other generalized series...

4. kinds and internal properties

   • common rational power e.g. sqrt(x)*(1 + x + x**2 + ...)
   • generalized series (Gruntz) with ... + c*x**p+ ... where c, p can be complex and rational.
   • generalized series on basis: 1 + log(x)*x + log(x)**x + ...

5. series (asymptotic extension) can have several variants of expansion for the same function ( and various interval of convergence)

6. series (as object) vs asymptotic expansion

   • asymptotic series expansion (with BigOh term, and precision)
   • more common and abstract symbolic expression as series itself.

7. We can permit transformation from one kind to another or not (extend==True)

8. tasks

• calculate series expansion of function (the same and for the abstract functions)
• construct generating functions from sequences or known coefficients
• construct and operate with series and asymptotic expansions (the same and for abstract function and operators)
• evalf (to calculate the series or coefficients)

Above sets can be collaborate together.

Collaboration with other topics or modules

1. Summation

2. polynomials

3. convergence