# How to denote a function that depends on $a_1, \ldots, a_n$ without the dots?

I know that I can wrote something like $$a_1 + \cdots + a_n$$ without the dots as $$\sum_{i=1}^n a_i$$ which seems clearer to me. As a programmer, I'd rather have a rule set with variables than something with dots where I have to extract the pattern from.

Is there some notation to do this for the parameters of a function? Say a Lagrangian like so: $$L\left(q_1, \ldots, q_n, \dot q_1, \ldots, \dot q_n, t\right)$$

The thought in the back of my head is the following. In Python, I could have a function like so:

f(x, y, z)


When I call that function, I could either to f(1, 2, 3) or I could do the following:

parameters = [1, 2, 3]
f(*parameters)


Where I basically “dump” that list of parameters into the parenteses of the function. Is there some math notation for the same thing?

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I think it's worth noting that even with $\sum_{i=1}^n$, there is a standard interpretation that is not really any more explicit than $a_1+\cdots+a_n$. – alex.jordan Oct 13 '12 at 17:57

I don't know if that's what you're looking for, but we do that last thing in $\mathbb R^n$ usually ; the vectors in $\mathbb R^n$ are defined as vectors of the form $$(x_1, \dots, x_n), \qquad x_i \in \mathbb R$$ but if we write $x = (x_1, \dots, x_n)$, when defining a function $f : \mathbb R^n \to \mathbb R$ for instance, we can just write $f(x)$ instead of $f(x_1, \dots,x_n)$. Is that what you were looking for?

For your Lagrangian for instance, you could define $q = (q_1, \dots, q_n)$, $\dot q = (\dot q_1, \dots, \dot q_n)$, and write $$L(q,\dot q, t)$$ instead of $$L(q_1, \dots, q_n, \dot q_1, \dots, \dot q_n, t).$$

Hope that helps,

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Well, $L\left(\vec{q}, \vec{\dot q}, t\right)$ seems clean and to the point. Defining vectors is probably the thing I am looking for! – Martin Ueding Oct 13 '12 at 17:27
@queueoverflow : I added something. Oh and you don't need to put arrows over the vectors. These are for children. =) – Patrick Da Silva Oct 13 '12 at 17:27
I see that the vector arrows are not used so much at university. They even write $\int \mathrm dx$ when I would write $\iiint \mathrm dV$ :-) – Martin Ueding Oct 13 '12 at 17:30
@queueoverflow : When one does measure theory he also often writes $\int_X f$. So yeah, arrows for children. Did I answer your question? – Patrick Da Silva Oct 13 '12 at 17:31
You answered my question, but I need to wait a little to accept it. – Martin Ueding Oct 13 '12 at 17:32

According to one professor of mine, $\underline{x} = x_1, \dots , x_n$.

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One professor of mine used the underscore for matrices. – Martin Ueding Oct 14 '12 at 18:53

You can write $\{x_i\}_{i\leq n}$ instead of $\{x_1, \dots, x_n\}$.

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True. But how can I make a set the parameters of a function? – Martin Ueding Oct 14 '12 at 14:29
queueoverflow: What's wrong with $f(\{x_i\}_{i\leq n})$? – F M Oct 14 '12 at 18:38
Maybe I am too much programmer here, but then $f$ is a function which domain is a set, not a bunch of variables. – Martin Ueding Oct 14 '12 at 18:47
Then redefine your function's domain to $n$-uples. IMO you shouldn't emphasize so much on this; after all it's just notation. Humans are not computers, as long as your notation isn't ambiguous there shouldn't be any problem. – F M Oct 14 '12 at 19:09
$n$-tuples $\neq$ sets. The order of the elements matters in $n$-tuples. – user26486 Nov 20 '14 at 21:44