All the definitions of limit superior and limit inferior I have seen (even in the books about complex analysis) define them for a real sequence only.

What could stop us from defining it as follow for a complex sequence?

$$\limsup\limits_{n\to\infty} z_n := \lim_{n\to\infty}\Big(\sup\{|z_k|:k \geq n\}\Big)$$ $$\liminf\limits_{n\to\infty} z_n := \lim_{n\to\infty}\Big(\inf\{|z_k|:k \geq n\}\Big)$$

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    $\begingroup$ Nothing is stopping you (but note that all you're doing is defining the limsup of $z_n$ to be the limsup of the real sequence $|z_n|$. Perhaps the more important question is, what might motivate this definition, beyond generalization for its own sake? $\endgroup$
    – Neal
    Jan 19 '13 at 0:00
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    $\begingroup$ Since the complex numbers do not form an ordered field, it doesn't really make sense to talk about sup or inf or limsup or liminf of complex sets or sequences. $\endgroup$
    – mrf
    Jan 19 '13 at 0:17

This feels like a poor generalization of these properties. One of the most valuable properties of the lim sup and lim inf is: 1) they always exist and 2) when they're equal, the sequence is convergent and converges to the limsup and liminf.

Worse, they don't even generalize, in the sense that we'd have for the constant real sequence $a_n = -1$ that $\limsup a_n = 1$ and $\liminf a_n = 1$, which is clearly not the limit of $a_n$.

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    $\begingroup$ Indeed. So what would be the correct generalization? $\endgroup$
    – Leo
    Jan 19 '13 at 0:09
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    $\begingroup$ @Ruslan: There isn't one. This is why we don't normally define limsup and liminf for complex sequences. $\endgroup$ Jan 19 '13 at 1:21

Those notions, as far as I can tell, cannot be smoothly generalized to the complex plane in an interesting way.

However, if you get a kick out of it, I think the generalization you're looking for is:

$$\limsup z_n = \sup \limits_{||u||=1} \limsup \pi_u(z_n)$$

Where $u\in \mathbb{R}^2$, and $\pi_u$ is the projection of a complex point on the linex generated by $u$, that is, if you think of a complex number $z$ as a vector in $\mathbb{R}^2$, $\pi_u(z) = u\cdot z$.

Note that in this case, since you're looking at projections on lines in all directions, $\liminf z_n = -\limsup z_n$. You could limit lines to be "left to right" and eliminate this problem (and then it would be a true generalization of the real case), but this seems rather arbitrary, and won't be, for example, a generalization of the "pure imaginary case".

What I would define if you want a more useful general notion, is something like "diameter of limit", akin to $\limsup x_n-\liminf x_n$ in real sequences, which could be defined as the supremum of distances between limits of $(z_n)$, or in a very similar manner to the definition above using projections as

$$\lim \mbox{diam } z_n = \sup \limits_{||u||=1} \left( \limsup \pi_u(z_n) - \liminf \pi_u(z_n)\right)$$

This is a more geometrically intersting quantity - it's a true gerneralization of $\lim \mbox{diam }$ as would be defined on reals and when it's $0$, the sequence converges (because it converges in all directions).


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