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Edit: This question is wrong. Ignore it. I've already flagged to request for deletion. Please just go to that question: What exactly are the differences between real multivariable limits and complex limits?


Say you want to disprove the existence of either of the ff $\lim_{z \to 0} \frac{Re(z)}{|z|^2}$, $\lim_{z \to 0} \frac{Im(z)}{|z|^2}$ (as in here). It seems we just change the limits to $\lim_{(x,y) \to (0,0)} \frac{x \ \text{or} \ y}{x^2+y^2}$ and then go about this calc2 way.

  1. So the rule is that complex limit doesn't exist if the $\mathbb R^2$ limit doesn't exist? In general, for $$\lim_{z \to z_0}[u(z)+iv(z)] \ \text{vs} \ \lim_{(x,y) \to (x_0,y_0)}[u(x,y)+iv(x,y)],$$

where $u$ and $v$ are real functions, is it that the LHS doesn't exist if the RHS doesn't exist?

  • Here, the RHS equals by theorem or by definition to $$\lim_{(x,y) \to (x_0,y_0)}u(x,y)+i\lim_{(x,y) \to (x_0,y_0)}v(x,y)$$
  1. BUT if the RHS exists, the LHS may or may not exist, i.e. existence of real limit is necessary but not sufficient for existence of complex limit? Please provide examples.

Note: For now, I'll just say the real functions $u,v$ without specifying specific domains and hope the above makes sense. If need be, then I can edit this question to be more specific about $u,v,z_0$, etc.


Related questions:

I've found several questions that talk about the relationship of complex derivative and real derivative, but what I'm not quite seeing is the general case/concept of complex vs real limits.

Differences between the complex derivative and the multivariable derivative.

Difference between the properties of differentiation in $\mathbb{C}$ and $\mathbb{R}^2$

Scalar field Derivative. Real vs Complex

Limit defintion of a function that is $\mathbb{R}$-differentiable but not $\mathbb{C}$ differentiable.

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    $\begingroup$ This question is a little odd in that often the definitions for complex limit and multivariable limit are often close enough for comparison, but you don't mention a definition for either in your post. Can you look them up (either in texts or online) and write about where you're getting stuck comparing and/or contrasting them? $\endgroup$
    – Mark S.
    Oct 5 '21 at 1:36
  • $\begingroup$ @MarkS. Maybe I wasn't so clear. I'm not (yet) asking how $\lim_{(x,y) \to (x_0,y_0)} f(x,y) = L$ differs from $\lim_{z=(x,y) \to z_0=(x_0,y_0)} u(x,y) + iv(x,y) = M$ (the answer is something about $(x,y)$ has to be in domain set in former case but not latter case or something.) My question is for the latter kind of limits, how do the ...oh wait never mind i was confused. yeah you're right question is odd. i was thinking of something else. actually LHS = RHS period. $\endgroup$ Oct 5 '21 at 4:33
  • $\begingroup$ @MarkS. here's the correct version of what i intended to ask: math.stackexchange.com/questions/4268942/… $\endgroup$ Oct 6 '21 at 1:01
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    $\begingroup$ Please do not vandalize your question. $\endgroup$
    – Xander Henderson
    Oct 14 '21 at 23:35
  • $\begingroup$ @XanderHenderson there is no question supposedly. i asked the wrong thing. i don't think this will benefit anyone to keep this question. i think it will just confuse people. $\endgroup$ Oct 21 '21 at 10:29
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Complex differentiation for $\mathbb C$ is defined differently than multivariate differentiation for $\mathbb R^2$. This is because you can divide by a complex number but not by a vector. It is NOT because limits work differently in the two spaces. In fact limits work identically in $\mathbb C$ and $\mathbb R^2$. More explicity, it is true that

$$ x_n + y_n i \to x + y i $$

as $n \to \infty$ if and only if

$$ \begin{pmatrix} x_n \\ y_n \end{pmatrix} \to \begin{pmatrix} x \\ y \end{pmatrix} $$

as $n \to \infty$.

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