Mathematicians & Big O, Big Omega: Usage & Context

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Discussion Overview

The discussion revolves around the usage and context of Big O, Big Omega, and related notations in mathematics, particularly in relation to asymptotic analysis and their application in expressing mathematical functions like sine.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants indicate that mathematicians use Big O and related notations primarily in asymptotic analysis.
  • One participant mentions the use of these notations to demonstrate the truncation of Taylor series, providing an example with the sine function.
  • Another participant suggests that the expression for sin(x) using Big O reflects its asymptotic behavior as x approaches zero, while noting that using an explicit remainder term would change the context.
  • There is a reiteration that the different perspectives on expressing sin(x) represent alternative ways of thinking about the function's behavior.

Areas of Agreement / Disagreement

Participants generally agree on the relevance of Big O and related notations in asymptotic analysis, but there are nuances in how they apply these concepts, particularly regarding the expression of functions and the implications of using explicit remainder terms.

Contextual Notes

Some assumptions about the definitions and contexts of Big O and related notations may be implicit, and there is an unresolved discussion about the implications of using different forms of expression for functions.

Swapnil
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Hi, I was wondering, do mathematicians (like computer scientists) use things like Big O, Big Omega, Little O, etc. a lot? If so, in what context?
 
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In asymptotic analysis.
 
Yep, asymptotic analysis accounts for most of it. It's also used to show the truncation of a (Taylor series) polynomial:

[tex]\sin(x)=x-\frac{x^3}{6}+O(x^5)[/tex]
 
And isn't that, really, an expression for sin(x)'s asymptotic behaviour as x ambles peacefully off towards the origin?:wink:

(If you write sin(x) with an explicit remainder term, say, by utilization of the mean-value theorem for integrals, then it is of course something different, but we wouldn't use O's in that case).
 
Last edited:
arildno said:
And isn't that, really, an expression for sin(x)'s asymptotic behaviour as x ambles peacefully off towards the origin?:wink:

Oh yes absolutely. It's just a different way of thinking about it.
 

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