# 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)`

• 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

##### Clone this wiki locally
You can’t perform that action at this time.
Press h to open a hovercard with more details.