How can I compute this limit: $$\lim_{z\rightarrow i} (z\cdot \overline{z})$$ (where $\overline{z}$ is the conjugate of $z$)?
The limit of $z\cdot\overline{z}$ as $z\to i$
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complex-analysis
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0Please make your titles informative, and please make the body of your post self-contained. In particular, since you are asking a question, the body of the message should actually contain a question. – 2011-02-16
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0Sory, I'll try that next time. – 2011-02-16
2 Answers
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As $\mathbb{C}$ comes nicely equipped with a metric, limits behave as nice as ever. Therefore, the limit of the product is the product of the limits. And the limit of the conjugate is the conjugate of the limit (this can be proved using a standard epsilon-delta argument). Hence, in your example: $$\lim_{z \to i} (z \cdot \overline{z})=\lim_{z \to i}(z) \lim_{z \to i}(\overline{z})= i \cdot(-i) = 1.$$
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0Thanks, I should revise the properties of the limit in the complex plane – 2011-02-16
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Hint: Is $z'$ the conjugate of $z$ and the lower dot a multiplication? If so, what is the problem? You might try expressing $z=a+bi$, with $a,b$ real and trying a real limit.
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0It is a conjugate, but I don't remember the symbol in LaTeX so I used that instead. Anyway, I'll try that. Thanks! – 2011-02-16
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0I am used to $\overline{z}$ but as people are used to different things it is worth defining. – 2011-02-16
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0I'm also used to that, but again, I didn't remember what's the command therefore I used $'$. – 2011-02-16
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0`\overline`, just got it! – 2011-02-16