= Recovering an analytic derivative from the boundary real part
{title2=$f'(0)=(\pi r)^{-1}\int_0^{2\pi}\operatorname{Re}f(re^{i\theta})e^{-i\theta}\,d\theta$}
For a <holomorphic function> on a neighbourhood of a closed disc, parametrize the <Cauchy derivative formula> on its boundary. The conjugate <Cauchy integral theorem> gives $\int\overline{f(re^{i\theta})}e^{-i\theta}\,d\theta=0$. Replacing $f$ in the derivative integral by $f+\bar f=2\operatorname{Re}f$ therefore gives the displayed identity. The boundary real part determines every nonconstant analytic coefficient; only an imaginary additive constant is lost.
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