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Is there any representation to state that a variable is close to (not equal) zero? Let me give you an example. Consider the function

$u(x)=\alpha (e^{i\omega \delta t}-1) f(x)$

I am interested in the function $u(x)$ when $\delta t$ is very small. For this case, it should be easy to see that

$u(x)|_{\text{small }\delta t} \approx i \alpha \omega \delta t f(x)$

Is there any "nice" notation to represent such an equation (without having to write small)? I thought that I could use the limit notation for that [for example, $\lim_{\delta t \to 0} u(t)$], but then I realized that if $\delta t$ goes to zero, then $u(x)=0$. Therefore, it is not what I need.

I found this link in the same forum, but it did not help.

Especially Lime
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jeb
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2 Answers2

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I think you are asking about the first order (linear) approximation using the derivative.

You might say $$ u(x)= i \alpha \omega \delta t f(x) + o(\delta t ). $$

The fact that this is an approximation for small $\delta t$ should be clear to your reader. If not you can say so.

Ethan Bolker
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  • I like to "o" notation, I have to say. What would you recommend me to include in my text or even in front of the $u(x)$ term to make it clear that $\delta t$ is small? My question is based on the fact that I will need to manage this equation further, and if I include the "o" notation, they will become too overloaded. The best that I could think of was something like $u(x)|_{\delta t = \epsilon}$, such that $o(\epsilon)$ is insignificant (or can be disregarded). Thank you. – jeb Mar 23 '21 at 18:41
  • The insignifance to first order is implicit in the use of the $o$ notation. You can just carry the $o(\delta t)$ term along in future calculations and throw it away at the end (r sooner) as appropriate. You could add "as $\delta t \to 0$" when you first state the equation. I would not want to read an invented notation like the one you propose. – Ethan Bolker Mar 23 '21 at 19:01
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The issue with taking limits is that as $\delta t\to 0$ both sides go to zero, but what you want to say is stronger than that.
I imagine what you actually mean by $u(x)\approx i\alpha\omega\delta tf(x)$ can be formalised as $$\lim_{\delta t\to 0}\frac{u(x)}{\delta t}=i\alpha\omega f(x).$$

This is simply a matter of rearranging so that the limit captures the relative error of the approximation, not just the absolute error.

Especially Lime
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