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

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Ukrainian Mathematical Journal Aims and scope

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. Mechnikov Odessa National University, Odessa, Ukraine

    L. L. Bezkorovaina

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  1. L. L. Bezkorovaina
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Correspondence to L. L. Bezkorovaina.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 5, pp. 596–609, May, 2019.

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Bezkorovaina, L.L. New Criterion for the Analyticity of Functions: Representation Via the Metric Tensors of the Surfaces Z = u and Z = 𝜐. Ukr Math J 71, 677–691 (2019). https://doi.org/10.1007/s11253-019-01670-3

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  • DOI: https://doi.org/10.1007/s11253-019-01670-3

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