generated from smyalygames/quartz
22 lines
622 B
Markdown
22 lines
622 B
Markdown
# Definition
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Let $z = x + iy$. Then its **complex conjugate** is
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$$\bar{z} := x-iy.$$
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# Properties
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The following hold:
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1. $\mid z \mid^2 = z \bar{z}$, ($=r^2$)
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2. $z + \bar{z} = 2 \mathrm{Re} z$,
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3. $z - \bar{z} = 2i \mathrm{Im} z$,
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4. $\overline{r e^{i \phi}} = r e^{-i \phi}$.
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> [!note]+ Note that (4) implies that
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>
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> $\mid \bar{z} \mid = \mid z \mid$.
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>
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> Also write that $z = r e^{i \phi}$ and $z' = r' e^{i \phi'}$.
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>
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> Then $zz' = rr'e^{i(\phi + \phi')}$. Then (1) implies that
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>
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> $\mid z z' \mid = rr' = \mid z \mid \mid z' \mid$.
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>
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> (Nice interplay between complex multiplication with absolute values).
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