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I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write

$$exp(\bar{z})=\overline{exp(z)}$$

the bar on the right is long and span throughout the whole 3 letters and z, what does this mean? And what do they mean by establishing the relation, do I show that they equal each other? There are no further explanations, they just said left for readers.

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The bar is the complex conjugate: $\overline{x+iy} = x - iy$. Your book is saying that for $z \in \mathbb{C}$, then the exponential of z-conjugate is the conjugate of the exponential of z.

Recall Euler's formula: $e^{iy} = \cos(y) + i \sin(y)$. So, $e^{x+iy} = e^x e^{iy} = e^x(\cos(y) + i \sin(y))$. Now, consider $e^{x-iy} = e^x(\cos(-y) + i \sin(-y))$. Sine is an odd function and cosine is an even function, so this is equivalent to $e^x(\cos(y) - i \sin(y))$, which is the complex conjugate of $e^x(\cos(y) + i \sin(y)) = e^{x+iy}$, Q.E.D.

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  • $\begingroup$ I understand now, thank you :) $\endgroup$
    – Akaichan
    Feb 17, 2014 at 2:44
  • $\begingroup$ Do you know what they mean by $\overline{sin(z)}$ then? What does the conjugate sine function look like? $\endgroup$
    – Akaichan
    Feb 17, 2014 at 5:03

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