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Is it true that for every $z \in \mathbb{C}$

$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z \end{align}$

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    It was a step in a solution I was trying to follow. The question itself was about proving a function to be harmonic and finding it's conjugate. Thank you.2012-02-20

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This is true because the coefficients of the power series expansion of these entire functions are real.