# Proof that an analytic function that takes on real values on the boundary of a circle is a real [duplicate]

I'm having trouble proving that an analytic function that takes on only real values on the boundary of a circle is a real constant. I started by writing

$f(r, \theta) = u(r, \theta) + i v(r, \theta)$

By definition, $v(r, \theta ) = 0$, so $\frac{d}{d\theta} v = 0$, and in fact, the nth derivative of $v(r,\theta)$ with respect to $\theta$ is 0. The Cauchy Riemann equations in polar coordinates imply that $\frac{d}{ dr} u(r, \theta ) = 0$

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