Symbol for "probably equal to" (barring pathology)?

I am writing lecture notes for an applied statistical mechanics course and often need to express the notion that something is very probably true for functional forms found in the wild, without launching into a full digression for pathological exceptions.

For example, I would like to remind students that a function's Taylor series sometimes has the useful property of converging to the function's value.

Is there a mathematical symbol for this type of "equality" that is more dignified than my current options: $$ f(x) \stackrel{\textrm{(good odds)}}{=} \sum_{k=0}^{\infty} \frac{(x-a)^k}{k!} f^{(k)} (a)$$ or $$ f(x) \texttt{ ¯\_(ツ)_/¯ } \sum_{k=0}^{\infty} \frac{(x-a)^k}{k!} f^{(k)} (a)$$

Solution 1:

I recommend handling this issue simply, such as:

1) "Under mild conditions we get from Fourier theory:"

$$ f(x) = \sum_{k=0}^{\infty} a_k \cos(2\pi k x) $$

2) "Under mild conditions we get from Taylor series theory:"

$$ f(x) = \sum_{k=0}^{\infty} a_k (x-a)^k $$

It is understood that the precise conditions under which equality holds can be obtained by looking at the details of that theory.

I would avoid:

-Unclear notation or phrases.
-Unclear, loaded, or advanced terminology.

For example: It sounds like you are talking about a class of problems for which no probability model is defined (or relevant). Thus, unclear phrases like "strong odds" will confuse. Funky equality signs will evoke laughter, but it will be nervous laughter since nobody will know what you are talking about. Advanced measure theory concepts such as "almost surely" will drive your teaching ratings down ("The professor expects us to know advanced probability which is not a prerequisite for this course...")

Solution 2:

When I’m writing up tentative results that remain yet to prove (perhaps subject to some additional regularity conditions or assumptions), I usually do this: $$x\,\overset{?}{{=}}\,y+z.$$

On a less serious note:

Let $f:\mathbb R\to\mathbb R$ be a function of class $\mathcal C^{\infty}$. Then, one has, for any $x\in\mathbb R$ and $a\in\mathbb R$, that $$f(x)\overset{*}=\sum_{k=0}^{\infty} \frac{(x-a)^k}{k!} f^{(k)} (a).$$ $\scriptsize{^{*}\text{Terms and conditions apply.}}$

Solution 3:

How about $f(x)\quad\overline{--}\quad etc.$ , indicating that there is a hole in the floor that you have to occasionally be careful not the step through?

Edit: better interpretation: call it "leaky equals", because occasionally the equality leaks out of it.