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Why do we flip the signs of all $i$ s in a complex number when we want to take the conjugate of it?

I mean, conjugating means making $x + iy$ into $x - iy$, but given a number of the form: $$\frac {x+iy}{x-iy}$$ or $$x+iy+e^{iz}$$ or any other form of complex number, why does flipping signs always work?

It works even when taking the complex conjugate of Schrodinger's wave equation. Is there a reason why any complex number, irrespective of their structure can be conjugated by flipping the sign of all the $i$ s (given that the conjugate exists) ?

Sidd
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2 Answers2

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Conjugation is an automorphism of the field of complex numbers. This just means that $\overline{z+w} = \overline{z} + \overline{w}$ and $\overline{z\cdot w} = \overline{z} \cdot \overline{w}$. In particular, $\overline{a\cdot z^n} = \overline{a} \cdot {\overline{z}}^n$. Any function expressible as a power series with real coefficients will therefore satisfy $\overline{f(z)} = f(\overline{z})$. In particular, this is the case for $f(z) = \exp z$ since $$ \exp z = \sum_{n=0}^\infty \frac{z^n}{n!}. $$ The coefficients $1/n!$ are all real, and so $$ \overline{\exp z} = \sum_{n=0}^\infty \overline{\frac{z^n}{n!}} = \sum_{n=0}^\infty \frac{\overline{z^n}}{n!} = \sum_{n=0}^\infty \frac{{\overline{z}}^n}{n!} = \exp \overline{z}. $$ In particular, $\overline{x+iy+e^{iz}} = \overline{x}-i\overline{y}+e^{-i\overline{z}}$.

Yuval Filmus
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    Just a small quibble. In order to extend an automorphism to convergent infinite sums we need to verify that the automorphism is continuous. It's not automatic. In this case conjugation is continuous so applying it to power series is ok. – user4894 Jan 23 '15 at 07:14
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It's because complex conjugation is an automorphism of $\mathbb{C}$. In other words for all $z, w \in \mathbb{C}$ we have $\overline{zw} = \overline{z} \ \overline{w}$ and $\overline{z + w} = \overline{z} + \overline{w}$. In addition, conjugation fixes the reals pointwise: if $r \in \mathbb{R}$ then $\overline{r} = r$. Your observation follows from those properties.

user4894
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