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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>
This follows from the previous theorem because:
* the Poisson kernel is the real part of the integrand above
* the real part of a holomorphic function is harmonic and determined by the holomorphic function up to a scalar
* the above formula defines a holomorphic function, the real part of which is given by the previous theorem
==Carathéodory's positivity criterion for holomorphic functions==
Let
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