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I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:

Suppose that $f(z)$ is analytic on the equilateral triangle, $T$ , with vertices at 1, $\omega$, $\omega^2$, where $$\omega=\exp{\frac{2\pi i}{3}}.$$ Then $$\iint \limits_T f^{\prime \prime}(z)\,dx\,dy=\frac{\sqrt{3}}{2}[f(1)+ \omega f(\omega)+\omega^2 f(\omega^2)].$$

A variant of the proof, as well as the generalization to other polygons, can be found in http://www.jstor.org/discover/10.2307/2002943?uid=3738936&uid=2&uid=4&sid=21104022760067 (Triangle Formulas in the Complex Plane, by Philip J. Davis).

Can the Grunsky-Motzkin-Schoenberg formula be generalized to other hypercomplex numbers (especially to quaternions and octonions)?

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  • $\begingroup$ gamarjobat zurab qartveli xart? $\endgroup$ – dato datuashvili May 15 '14 at 4:59
  • $\begingroup$ gagimarjot, ki qartveli var. $\endgroup$ – Zurab Silagadze May 15 '14 at 5:08
  • $\begingroup$ sasiamovnoa ,am saitze ishviatad vpostav xolme shekitxvebs ufro am saitze shevdivar xolme math.stackexchange.com/questions/794713/… igive shekitxva davposte aq da imedia pasuxs gamcemen, $\endgroup$ – dato datuashvili May 15 '14 at 5:10
  • $\begingroup$ dato, darcmunebuli ara var rom gavige tkveni shekitxva (tanac specialisti ara var am sakmeshi), magram mivutite erti statia romelic imedia sheijleba gamogadget. ecadet kitxva ufro daxvecot (inglisuric), rom ufro nateli da gasagebi ikos. gisurvebt carmatebas tvens profesiashi. $\endgroup$ – Zurab Silagadze May 15 '14 at 6:01
  • $\begingroup$ madlobt gaixaret, shekitxva exeba chemi sadisertacio temas,kerdzod perioduli deterministuli komponentebi rodesac tetri xmauris zegavlenis qvehsh aris,am dros shemtxveviti procesis(stochasturi processis) tipi mainteresebda,misi statistikuri ganawilebis forma $\endgroup$ – dato datuashvili May 15 '14 at 6:08

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