As in the title I would like to find a holomorphic function $f$ such that $\Re f(z)=(\Re z)^2$ for every $z$ on the unit circle.
2026-03-28 07:36:50.1774683410
Holomorphic function with real part being $(\Re z)^2$ on unit circle
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If $x^2+y^2=1,$ then $$x^2 = \text {Re}\left( \frac {(x+iy)^2+1}{2}\right).$$ Thus $f(z)= \dfrac {z^2+1}{2}$ is a solution.