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| In [[complex analysis]], a branch of mathematics, the '''Schwarz integral formula''', named after [[Hermann Schwarz]], allows one to recover a [[holomorphic function]], [[up to]] an imaginary constant, from the boundary values of its real part.
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| ==Unit disc==
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| Let ''ƒ'' = ''u'' + ''iv'' be a function which is holomorphic on the closed unit disc {''z'' ∈ '''C''' | |''z''| ≤ 1}. Then
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| : <math> f(z) = \frac{1}{2\pi i} \oint_{|\zeta| = 1} \frac{\zeta + z}{\zeta - z} \text{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta}
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| + i\text{Im}(f(0))</math>
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| for all |''z''| < 1. | |
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| ==Upper half-plane==
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| Let ''ƒ'' = ''u'' + ''iv'' be a function that is holomorphic on the closed [[upper half-plane]] {''z'' ∈ '''C''' | Im(''z'') ≥ 0} such that, for some ''α'' > 0, |''z''<sup>''α''</sup> ''ƒ''(''z'')| is bounded on the closed upper half-plane. Then
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| : <math>
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| f(z)
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| =
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| \frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} \, d\zeta
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| =
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| \frac{1}{\pi i} \int_{-\infty}^\infty \frac{Re(f)(\zeta+0i)}{\zeta - z} \, d\zeta
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| </math>
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| for all Im(''z'') > 0.
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| Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.
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| == Corollary of Poisson integral formula ==
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| The formula follows from [[Poisson integral formula]] applied to ''u'':<ref>
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| {{cite web
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| |url=http://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE
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| |title=Lectures on Entire Functions - Google Book Search
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| |publisher=books.google.com
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| |accessdate=2008-06-26
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| |last=
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| |first=
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| }}
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| </ref><ref>The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html</ref>
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| : <math>u(z) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} \, d\psi\text{ for }|z| < 1.</math> | |
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| By means of conformal maps, the formula can be generalized to any simply connected open set.
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| == Notes and references ==
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| <references />
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| * [[Lars Ahlfors|Ahlfors, Lars V.]] (1979), ''Complex Analysis'', Third Edition, McGraw-Hill, ISBN 0-07-085008-9
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| * Remmert, Reinhold (1990), ''Theory of Complex Functions'', Second Edition, Springer, ISBN 0-387-97195-5
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| * Saff, E. B., and A. D. Snider (1993), ''Fundamentals of Complex Analysis for Mathematics, Science, and Engineering'', Second Edition, Prentice Hall, ISBN 0-13-327461-6
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| [[Category:Complex analysis]]
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