Question regarding notation when omitting terms

In summary, the conversation discusses expanding a function up to first order around a specific point. The notation used for this is ##f(x) = f(0) + (df/dx)\Big |_0 \; x + \mathcal O(x^2)##. However, if one wants to expand the function in the whole series and only show the first order term in x, the notation used is ##f(x) = f(0) + (df/dx)\Big |_0 \; x + \sum \frac{d^{k-2}f}{dx^{k-2}}x^{k-2} / (k-2)!##, which can also be represented as ##f(x) =
  • #1
kent davidge
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If we want to expand a function ##f(x)## up to first order around ##x = 0## say, we usually write ##f(x) = f(0) + (df/dx)|_0 x + \mathcal O(x^2)##.

But what if I want to expand ##f(x)## in the whole series, and showing only the first order term in x? What notation do you use for that? (Aside from ##f(x) = f(0) + (df/dx)|_0 x + \sum \frac{d^{k-2}f}{dx^{k-2}}x^{k-2} / (k-2)!##.)

My thought: ##f(x) = f(0) + (df/dx)|_0 x + \mathcal O(x^{\geq 2})##
 
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  • #2
Hi,
kent davidge said:
if I want to expand ##f(x)## in the whole series
You don't want that, because it doesn't make sense. The only term of interest around zero is the ##x^2## term. All higher order terms in ##f(x) - f(0) - (df/dx)\Big |_0 \; x \ ## vanish for ##x\downarrow 0##.

In other words, ##\mathcal O(x^{2}) ## is the same as your ##\mathcal O(x^{\geq 2})## (or: implies it).
 
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  • #3
x3 < x2 for small x, therefore ##ax^3 \in O(x^2)##, same for all higher orders.
 
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  • #4
mfb said:
x3 < x2 for small x, therefore ##ax^3 \in O(x^2)##, same for all higher orders.
I'm not considering only small ##x## though
 
  • #5
Doesn't matter. The meaning doesn't change:
$$f(x) - f(0) - (df/dx)\Big |_0 \; x \ = \mathcal O (x^2) \equiv \lim_{x \downarrow 0 }{f(x) - f(0) - (df/dx)\Big |_0 \; x \over x^2} \ \ \text {is finite}$$no more, no less. If you want to consider huge ##x## is upon you -- it probably won't be a good approximation, though.

1575396807986.png
( from wikipedia sin )
 
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  • #6
For large x you have to choose between physics and computer science convention of the notation, but in the latter case considering the constant and linear term is irrelevant and you'll likely need a completely new approach.
 
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1. What is notation when omitting terms?

Notation when omitting terms refers to the practice of leaving out certain parts of a mathematical expression or equation, usually for the sake of simplicity or brevity. It is commonly used in calculus and other branches of mathematics.

2. When should I use notation when omitting terms?

Notation when omitting terms should be used when the omitted terms are either insignificant or do not affect the overall result of the equation. It is also useful when the omitted terms are well understood and can be easily inferred by the reader.

3. How do I denote omitted terms?

Omitted terms are typically denoted by using an ellipsis (three dots) or a dash. The specific notation used may vary depending on the context and the preference of the author. It is important to clearly indicate that terms have been omitted to avoid confusion.

4. Can I omit terms in any mathematical expression?

Yes, you can omit terms in any mathematical expression as long as it does not change the meaning or result of the equation. However, it is important to use proper notation and to make sure that the omitted terms are not essential to understanding the equation.

5. Are there any rules or guidelines for using notation when omitting terms?

There are no strict rules for using notation when omitting terms, but there are some general guidelines to follow. It is important to use notation consistently throughout a document and to clearly indicate which terms have been omitted. Additionally, it is best to only omit terms that are well understood and do not significantly impact the overall equation.

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