How would someone go about writing a general expression for a function of a single variable and a constant? For example, if we have a function of two variables 'x' and 'y' we can use f(x,y). If we had a function of 'x' and the constant '∏' is it acceptable to write it as f(x,π)?
Why would you want to do that? By definition, a constant does not change, so it should not influence the dependent variable. Functions like ##x \rightarrow \pi x## or ##x \rightarrow \sin \pi x## are completely adequately represented as ##f(x)## (omitting the ##\pi##).
This is in reference to new functions where the constant, π for this particular example, is relevant. Proper notation is paramount otherwise there is no point. On that note, and considering the confusion about the example f(x,∏), this particular representation does not seem appropriate, do you have an alternative suggestion?
There are lots of notations for a function ##f## in variable ##x## with some constant parameter ##a##, besides just plain old ##f(x, a).## Some other examples are ##f_a(x)##, ##f^{(a)}(x)##, ##f(x; a).##
Excellent, thank you olivermsun. Do any of those have a particular meaning or is, ##f(x, a)## = ##f_a(x)## = ##f^{(a)}(x)## = ##f(x; a).## for all cases? Does anyone else have anything to add?
It depends on the context it's used in. Use a notation that doesn't cause confusion in your context and be consistent.
I have seen the notation f(x; c) used to indicate a function of the variable x which depends upon the parameter c. I think the word "parameter" is better here than "constant". (You understand that the distinction between a "variable" and a "constant" that can take on different values is pretty slim! f(x; c) is understood to mean a family of functions of x, each possible value of c giving a different function in that family.)
I have seen many examples of 'family of functions' (some even posted on PF) so this notation is good to know. On another note, for this particular instance I am interested in notation for a function of one variable that also happens to have a single known constant (e.g. π, or e, or phi, etc.) as part of that function. If I had a function of a single variable 'x' and constant '∏' is f(x,∏) an acceptable form to describe said function?
Only if there's no chance of confusion. For example, I think writing ##\log_e(x)## and ##\log_{10}(x)## is a good idea when either or both could be used.
I understand this sentiment, however if it were that simple I wouldn't be here. Do you think using the form f(x,some constant) is adequate?
In this case you would write f(x) = lnx =log_e x. You're still not writing the e in the name of the function, f. I prefer ## f_c(x) ## to indicate that the function is only a function of x.
You are. The ##e## appears in the subscript of the function name ##\log##. Anyway it's just one possible convention. The point is to distinguish between base ##e## and ##10## clearly.
Hm, yeah I suppose you're right about this one. Though a comment, even here you aren't using the notation ##f(x,a)##, it's still the notation ##f_a(x)##.