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Is there some notation in terms of $n,k$, I can use to represent the complex exponential $e^{2\pi i\frac{k}{n}}$, I find by writing the exponential out, I often make mistakes and it is timely to write out when I must express a large number of them, I don't want to make up my own notation because, I would rather be used to writing roots of unity in a way others understand.

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$\exp\left(2\pi i x\right)$ is sometimes written $e(x)$ (especially in a number theory setting, for instance with Gauss sums, in my experience), in which case what you have would be $e(k/n)$. –  anon Dec 13 '12 at 17:01
    
I like this notation a lot better then the other ones, but is it really commonly used? –  Ethan Dec 13 '12 at 17:05
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Dear Ethan, as @anon writes, for number theorists (especially analytic number theorists) the notation $e(x)$ is quite common. It may be much less so in other fields (though I don't know). Regards, –  Matt E Dec 13 '12 at 23:40

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You often see $\omega$ or $\omega_n$ used to denote $e^{2 \pi i/n}$, then it's quite simply $\omega_n^k$. I've also seen $\zeta$ and $\zeta_n$ used for this purpose.

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I'd say I've seen $\zeta_n$ far more often than $\omega_n$. (See most references on cyclotomic fields for instance.) Often $\omega$ is specifically used for $\zeta_3$, too. –  anon Dec 13 '12 at 17:02
    
@anon: In Galois theory and algebraic number theory [$\to$ cyclotomic fields] I've usually seen and used $\zeta_n$, but I've encountered $\omega$ lots in complex analysis $-$ I don't think I'm really qualified to say which is more common though! –  Clive Newstead Dec 13 '12 at 17:09
    
I see and use $\zeta$ most often, followed by $\omega$. I typically use $\zeta$ except for the third roots of $1$, which for some reason seem to be more commonly denoted by $\omega$. I've also seen $\varepsilon$ used for roots of $1$. –  Brett Frankel Dec 13 '12 at 18:08

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