# Is it okay to write $n = n(k)$ to say that $n$ is a function of $k$?

In Physics, there is the index of refraction $n$. Sometimes, it will be just a scalar, sometimes it will be a function of the wave number $k$. I often see $n = n(k)$ to denote that $n$ is actually a function of $k$. Since in Physics the function arguments are often omitted, this kind of works.

But I would think (maybe more from a programming perspective) that $n$ is a function then and $n(k)$ a scalar, the value of $n$ at $k$.

Would it be only correct to write $n \colon k \mapsto n(k)$ in such situtations?

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I think $n$ is written as a function of $k$ only when it is important to do so; if working with general properties of $n$ that does not depend on $k$ it is irrelevant. – Marra Apr 12 '13 at 11:47

This is somewhat of an overload of the letter $n$, but in some cases it is clear enough to be acceptable.

For example if $n(k)$ is some index, and we want to focus on one particular $k$ for the moment, and not carry the $n(k)$ around everywhere. In such cases it can be acceptable to write "Let $n=n(k)$, then $X_n$ bla bla bla".

A particular example is some family of subsets of $\Bbb N$, $\{A_n\mid n\in\Bbb N\}$. And for every $k$ we let $n(k)$ to be the least index such that $k\in A_{n(k)}$. Now we want to focus on what happens for a particular $k$, so we may write "Let $n=n(k)$, then $A_n$ ...".

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Okay, so in a very strict sense it is not okay. That is what I wanted to know. – Martin Ueding Apr 12 '13 at 11:57
Think about it in the context of real numbers for a moment, writing $f=f(x)$ and treating $f$ as a number makes no sense. This is the same issue here, only when it comes to indexing it is often well-understood where $n$ is a function and where it is the particular index of interest. – Asaf Karagila Apr 12 '13 at 11:59

It would probably be better to write $n_k = n(k)$ or something to that effect. If you consider the function as a map, it's more logical.

$n = n(k)$ makes me think that you mean "The value of $n(x)$ for any $x$ is equal to the value of $n(x)$ at $k$."

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