Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I - Gryshchuk, S. V. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
Among all two-dimensional algebras of the second rank with unity e over the field of complex numbers C, we find a semisimple algebra B_0 = { c_1e + c_2ω: c_k C, k = 1, 2} , ω^2 = e, containing bases (e_1, e_2), such that e^4_1 + 2pe^2_1e^2_2 + e^4_2 = 0 for every fixed p > 1. A domain { (e1, e2)} is described in the explicit form. We construct B_0 -valued “analytic” functions Φ such that their real-valued components satisfy the equation for the stress function u in the case of orthotropic plane deformations (^4x^4 + 2p ^4x^2 y^2 + ^4y^4) u(x, y) = 0, where x, y are real variables.