Is There a Computer Algebra System That Can Handle Big O Error Terms?

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SUMMARY

The discussion centers on the need for a computer algebra system (CAS) capable of manipulating big O error terms, specifically mentioning tools like Math'ca, Maple, and Maxima. Users have noted that while Mathematica includes an O function, it fails to handle expressions correctly, particularly in operations involving division and more complex functions like inverses and integrals. The community seeks a CAS that can accurately manage these terms, especially for limits approaching infinity, and questions the feasibility of developing such a system.

PREREQUISITES
  • Understanding of big O notation in mathematical analysis
  • Familiarity with computer algebra systems (CAS) such as Mathematica, Maple, or Maxima
  • Basic knowledge of calculus, particularly limits and integrals
  • Experience with symbolic computation and algebraic manipulation
NEXT STEPS
  • Research the capabilities of Maxima for handling big O terms
  • Explore advanced features of Maple for symbolic manipulation
  • Learn about Mathematica's limitations and potential workarounds for big O notation
  • Investigate the development of custom functions in CAS for handling complex limits
USEFUL FOR

This discussion is beneficial for mathematicians, computer scientists, and software developers interested in symbolic computation, particularly those working with asymptotic analysis and error term manipulations in mathematical expressions.

CRGreathouse
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I was wondering if there was a computer algebra system (like Math'ca, Maple, or Maxima) that could handle manipulations with big O error terms.

Mathematica has an O function, but it doesn't work properly:
Code: 
> x^2 Log[x] + x Log[x]^2 + x / Log[x] + O[x]

O[x^1]
(it should be "x^2 Log[x] + x Log[x]^2 + O[x]")

But I'm interested in more than just cutting off small terms. I'd like to be able to do division
Code: 
> (x^2 + 3)/(x^2 + O(x))

1 + O(1/x)

and harder functions like inverses and integrals
Code: 
> inverse(x^2 + O(x), x)

±sqrt(x) + O(1)

(Note: the Os in my examples are about +infinity; a system handling other limits would be nice.)

Is there any system that can do this? If not, is there a reason? Is it harder than it seems?
 

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