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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 [[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 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})={1\over 2\pi}\int_0^{2\pi} {1-r^2\over 1-2r\cos (\theta-\varphi) + r^2} \, d\mu(\varphi).</math>
==Herglotz representation theorem for holomorphic functions==
A holomorphic function ''f'' on the unit disk with ''f''(0) = 1 has positive real part if and only if there is a probability measure μ on the unit circle such that
:<math> f(z) ={1\over 2\pi} \int_0^{2\pi} {1 + e^{-i\theta}z\over 1 -e^{-i\theta}z} \, d\mu(\theta).</math>
==Carathéodory's positivity criterion for holomorphic functions==
Let
:<math> f(z)=1 + a_1 z + a_2 z^2 + \cdots</math>
be a holomorphic function on the unit disk. Then ''f''(''z'') has positive real part on the disk
if and only if
:<math> \sum_m\sum_n a_{m-n} \lambda_m\overline{\lambda_n} \ge 0</math>
for any complex numbers λ<sub>0</sub>, λ<sub>1</sub>, ..., λ<sub>''N''</sub>, where
:<math> a_0=2,\,\,\, a_{-m} =\overline{a_m}</math>
for ''m'' > 0.
==References==
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