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.

  • $\begingroup$ Really, what's wrong with $\delta t<<1$? $\endgroup$ Mar 23 at 15:44
  • $\begingroup$ $\delta t\ll 1$ is good $\endgroup$
    – K.defaoite
    Mar 23 at 15:55
  • $\begingroup$ I think it is okay. Nevertheless, I would need to use something like $|\delta t| << 1$ to make it clear that $\delta t$ is positive? $\endgroup$
    – jeb
    Mar 23 at 16:31

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.

  • $\begingroup$ 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. $\endgroup$
    – jeb
    Mar 23 at 18:41
  • $\begingroup$ 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. $\endgroup$ Mar 23 at 19:01

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.