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

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.