Notation issue - Asymptotic behaviour: is $\sim$ too restrictive?

Question

As a student I am completely unable to understand unambiguously what is meant by a notation such as $$f \sim g $$ when in Physics the behaviour of two functions at infinity is evaluated. I found a definition according to which the notation is equivalent to stating that $$ \lim_{x \to \infty} \frac {f}{g} = 1$$ but is this not too restrictive? This then means is not correct to say that $$x^2$$ and $$\frac{1}{2} x^2$$ have the same asymptotic behaviour at infinity. I find it puzzling. Thanks PS: The question is admittedly trivial but an extensive search left it unanswered, so I appeal to your patience...

Answer 1

I'm not 100% sure the following will answer your answer, but by definition, $f \sim g$ (implicitly, at $+\infty$) if $|f(x) - g(x)| = o(|f(x)|)$ (this is the little o notation). In other words, $$\forall \epsilon > 0, \exists x_0 \text{ s.t. } \forall x \ge x_0, |f(x) - g(x)| < \epsilon |f(x)|.$$

When $g(x)$ does not vanish, it is equivalent to saying that $$\lim_{x \to +\infty} \frac{f(x)}{g(x)} = 1$$ (exercise).

You are indeed correct that $x^2 \not\sim x^2/2$, because $1/2 \neq 1$. The notation $\sim$ is restrictive on purpose, and in some situations it is helpful to have this "strict" asymptotic equivalence.

For example if $f \sim g$, then you know that $\lim_{x \to +\infty} f(x) - g(x) = 0$, and so you can now look for a function (converging to zero) such that $f - g \sim h$, etc. It's called an asymptotic expansion (similar to a Taylor series). For example, if $H_n$ is the $n$th harmonic number, then $H_n \sim \ln n$, and then $H_n - \ln n \sim \gamma$, etc. If you didn't have this strict asymptotic comparison, it would be possible to have $\lim_{x \to \infty} f(x) - g(x) = +\infty$.

But there is another notation that's useful, $\Theta$: by definition, $f = \Theta(g)$⁽¹⁾ iff $f = O(g)$ and $g = O(f)$. In other words, there exist constants $A,B > 0$ such that for big enough $x$, $|f(x)| \le A |g(x)|$ and $|g(x)| \le B |f(x)|$. It is clear that $f \sim g \implies f = \Theta(g)$. But the reverse is not true, as for example $x^2 = \Theta(x^2/2)$.

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⁽¹⁾ Be careful, the equality is just a notation! $\Theta(g)$ is not a function, it's actually a set. $f = \Theta(g)$ is not an equality. It's like when you write $f = o(g)$. A more correct way to write this would be $f \in \Theta(g)$: "$f$ is in the set of functions which are asymptotically bounded above and below by $g$". For historical reasons, the notation $f = \Theta(g)$ keeps getting used (and in physics I doubt you will ever see the correct notation).

Answer 2

In some italian textbooks we use two different symbols: as $x \to x_0$

1. $f\sim g$ to mean that $\lim_{x \to x_0} \frac{f(x)}{g(x)}=\ell \in \mathbb{R} \setminus \{0\}$, and
2. $f \asymp g$ to mean that $\lim_{x \to x_0} \frac{f(x)}{g(x)}=1$.

This notation is not really popular, but it helps you to understand where a multiplicative constant is not enough.