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In [[mathematics]], a '''positive harmonic function''' on the [[unit disc]] in the [[complex numbers]] is characterized as the [[Poisson integral]] of a finite [[positive measure]] on the circle. This result, the ''Herglotz representation theorem'', was proved by [[Gustav Herglotz]] in 1911. It can be used to give a related formula and characterization for any [[holomorphic function]] on the unit disc with positive real part. Such functions had already been characterized in 1907 by [[Constantin Carathéodory]] in terms of the [[Positive definite function on a group|positive definiteness]] of their [[Taylor coefficient]]s.
==Herglotz representation theorem for harmonic functions==
A positive function ''f'' on the unit disk with ''f''(0) = 1 is harmonic if and only if there is a [[probability measure]] μ on the unit circle such that
:<math> f(re^{i\theta})=\int_0^{2\pi} {1-r^2\over 1-2r\cos (\theta-\varphi) + r^2} \, d\mu(\varphi).</math>
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