# Grunsky-Motzkin-Schoenberg formula

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)?

• gamarjobat zurab qartveli xart? – dato datuashvili May 15 '14 at 4:59
• gagimarjot, ki qartveli var. – Zurab Silagadze May 15 '14 at 5:08
• 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, – dato datuashvili May 15 '14 at 5:10
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• 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 – dato datuashvili May 15 '14 at 6:08