New Criterion for the Analyticity of Functions: Representation Via the Metric Tensors of the Surfaces Z = u and Z = 𝜐

  • L. L. BezkorovainaEmail author

We establish a new criterion for the analyticity of a function \( w=u+i\upupsilon \kern0.5em \mathrm{or}\kern0.5em \overline{w}=u-i\upupsilon, \) where u(x, y), v(x, y) ∈ C1(G) in a domain G. It is expressed via the metric tensors of the surfaces Z = u and Z = v : g11− a22 = 0, g12 + a12 = 0, and g22− a11 = 0. We also discover some other equivalents of the analytic function and establish the invariance of the obtained relations under conformal transformations. The generalized version of the new criterion is also proposed.


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Authors and Affiliations

  1. 1.Mechnikov Odessa National UniversityOdessaUkraine

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