Positive harmonic function: Difference between revisions

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Revision as of 00:48, 22 December 2011

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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