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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? $\endgroup$ May 15, 2014 at 4:59
  • $\begingroup$ gagimarjot, ki qartveli var. $\endgroup$ May 15, 2014 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, $\endgroup$ May 15, 2014 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. $\endgroup$ May 15, 2014 at 6:01
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    $\begingroup$ @Carlo Beenakker. Nothing mathematical in the above comments. Just two Georgians met and exchanged words. $\endgroup$ May 15, 2014 at 8:55

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