The Loewner differential equation describes a growing family of simply connected planar domains through ordinary differential equations for their normalized conformal maps.
For a capacity-parameterized locally growing hull family,where the continuous real function is the Loewner driving function.
The Loewner driving function is the real boundary point at which the mapped-out hull grows. Scaling the hulls by and time by transforms it to .
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The Loewner differential equation is a key equation in complex analysis, particularly in the study of conformal mappings and stochastic processes. It is named after the mathematician Charles Loewner, who introduced it in the context of the theory of univalent functions. The Loewner equation describes a continuous deformation of a conformal map defined on a complex plane.