New criterion for the analyticity of a function: representation via the metric tensors of the surfaces $Z = u, Z = v$

  • L. L. Bezkorovaina

Abstract

UDC 514.75:517.53
We establish a new criterion for the analyticity of a function $w=u+iv$ or $\overline{w}=u-iv,$ $u(x, y),$ $v(x, y) \in C^1(G)$ in the domain $G.$ It is expressed via the metric tensors of the surfaces $Z=u$ and $Z=v\colon g_{11}-a_{22}=0,$ $g_{12}+a_{12} =0,$ $g_{22}-a_{11}=0.$ 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.
Published
25.05.2019
How to Cite
BezkorovainaL. L. “New Criterion for the Analyticity of a Function: Representation via the Metric Tensors of the Surfaces $Z = U, Z = V$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 5, May 2019, pp. 596-09, http://umj.imath.kiev.ua/index.php/umj/article/view/1460.
Section
Research articles