Notes on variables

things to talk about eventually:

Concepts to discuss:

Mathematicians and computer scientists are usually not careful with a function versus the output of a function. So for instance when using the big-Oh notation, people will write \(\mathcal{O}(n)\) (which is imprecise, because it doesn’t specify what the input variable is; is \(n\) the parameter or a constant?) instead of “\(\mathcal{O}(f)\), where \(f(n) = n\)” or “\(\mathcal{O}(\lambda n.n)\)”.

Similarly, when dealing with Laplace transforms, it seems common to write both \(\mathcal{L}\{f(t)\} = F(s)\) and \(\mathcal{L}\{f\} = F(s)\); but \(s\) is not present on the left hand side of either denotation! To be pedantic, we would need to write \(\mathcal{L}\{f\} = F\) or \(\mathcal{L}\{f\}(s) = F(s)\) or \(\mathcal{L}\{\lambda t.f(t)\}(s) = F(s)\).

In differential equations, it also seems common to write something like \[y'' + p(t)y' + q(t)y = f(t)\] but \(y\) is a function depending on \(t\), so shouldn’t it instead be the following? \[y''(t) + p(t)y'(t) + q(t)y(t) = f(t)\] Or more simply \[y'' + py' + qy = f\]