If two complex numbers are close enough then their Arguments too?

Question

Let $z_1=r_1e^{i\theta_1}$ and $z_1=r_2e^{i\theta_2}$ be two complex numbers such that $-\pi<\theta_1,\,\theta_2\leq \pi$. My question is: If $|z_1-z_2|<\delta$, for some $\delta>0$ sufficiently small then $|\theta_1-\theta_2|<\lambda$ for some $\lambda$ sufficiently small ?

Any suggestion is welcome.

The argument is the angle between them. If both are close to the origin the angles need not be close. — fleablood, Feb 20 '21 at 19:51

score 3 · Accepted Answer · answered Feb 20 '21 at 19:40

3

In general, no. If $z_1=\frac\delta4$ and $z_2=-\frac\delta4$, then their arguments are $0$ and $\pi$ respectively, but $|z_1-z_2|=\frac\delta2<\delta.$

answered Feb 20 '21 at 19:40

José Carlos Santos

427,504

It's ok, but I'm forget that such complex numbers are not reals. – ElliptCg Feb 20 '21 at 19:46
I'm not sure that I have understood your remark. Do you think or not that what I wrote answers your question. Note that $\frac{\delta i}4$ and $-\frac{\delta i}4$ would also work. – José Carlos Santos Feb 20 '21 at 19:48
Oh!, yes now I understand. Thank you for your examples. – ElliptCg Feb 20 '21 at 19:52

If two complex numbers are close enough then their Arguments too?

1 Answers1