In my continued effort to optimize common patterns that I am seeing in engineering models, I’ve found a more general pattern of functions of the form:
$$ fm(x) = \frac{f(x)}{x} $$
where “fm” is an arbitrary naming that I have chosen for this post. My question for you all is:
Is there an existing generalized naming for that pattern that would be better than “fm” (using an "m" suffix)?
Or even: Is there any specific case that has a traditional name that I can then adopt as a general naming pattern?
For example, I often see this pattern:
$$ sinm(x) = \frac{sin(x)}{x} $$
where not only can I optimize $\frac{sin(x)}{x}$ for efficiency, but also it is well-defined as equal to $1$ at $x = 0$. So, does that have an existing name?
Similarly,
$$\begin{align} tanm(x) & = \frac{tan(x)}{x} \\\\ sinhm(x) & = \frac{sinh(x)}{x} \end{align} $$
are both well-defined as 1 at x == 0 and can be optimized.
Similar patterns I see occasionally are:
$$ \begin{align} expm(x) & = \frac{exp(x)}{x} = \frac{e^x}{x} \\\\ lnm(x) & = \frac{ln(x)}{x} \end{align} $$
I am also seeing these patterns quite often:
$$ \begin{align} quadraticm(x, a, b, c) & = \frac{quadratic(x, a, b, c)}{x} = a x + b + \frac{c}{x} \\\\ cubicm(x, a, b, c, d) & = \frac{cubic(x, a, b, c, d)}{x} = a x^2 + b x + c + \frac{d}{x} \end{align} $$
Using an “m” or “m1” suffix for “minus one order” is the best naming I’ve come up with so far. Another option is “d” for damped… though that would be linearly damped, as opposed to exponentially damped where I’d be dividing by $e^x$ (which may be another pattern of value).
So, what general naming pattern would you mathematicians find natural for $\frac{f(x)}{x}$?
I have to name them something in my software… just looking for some guidance from mathematicians on any existing or natural pattern that I should be following… as opposed to just picking something blindly that might be a poor choice.
Thanks for any recommendations!