From conformal mapping theory this quadratic map is known to change a half plane in the \ zeta-space into potential flow around a semi-infinite straight line.
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The conformal mapping property of analytic functions ( at points where the derivative is not zero ) gives rise to a geometric property of harmonic conjugates.
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All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings.
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Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries.
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His " Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale " ( 1882 ) treats complex analysis in a geometric way, connecting potential theory and conformal mappings.
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*PM : proof of criterion for conformal mapping of Riemannian spaces, id = 8505 new !-- WP guess : proof of criterion for conformal mapping of Riemannian spaces-- Status:
37.
*PM : proof of criterion for conformal mapping of Riemannian spaces, id = 8505 new !-- WP guess : proof of criterion for conformal mapping of Riemannian spaces-- Status:
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Just as the M�bius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping.
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To see this, it can be assumed that both domains contain the point " and that the conformal mapping " f " carries " onto ".
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Uniformization can also be proved using the Beltrami equation . constructed the conformal mapping to a circular domain by minimizing a nonlinear functional a method that generalized the Dirichlet principle.
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