New criterion for the analyticity of a function: representation via the metric tensors of the surfaces $Z = u, Z = v$
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...
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| Datum: | 2019 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1460 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | 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. |
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