Mathematics & Physics 2016, 9(4), 416–426 УДК 517.55 Three Families of Functions of Complexity One Valery K. Beloshapka∗ Department of Mechanics and Mathematics Moscow State University GSP-1, Moscow, 119991 Russia Received 17.06.2016, received in revised form 29.07.2016, accepted 24.08.2016 Three rare families of functions of analytic complexity one were studied. <...> Main results are the description of linear differential equations with solutions of complexity one (Theorem 2), the description of L-pairs of complexity one (Theorem 5), the description of O(2)-simple functions (Theorem 7). <...> A method of measuring the complexity of an analytic function in two variables, possibly multivalued, is proposed in [3]. <...> For any analytic function of two variables z(x, y) one can define its complexity N(z). <...> Linear equations with constant coefficients Consider the pair of functions (z1 = eax+by, z2 = epx+qy). <...> What condition on (a, b, p, q) provides that the complexity of all linear combinations of z1 and z2 does not exceed one? <...> All rights reserved c – 416 – Valery K.Beloshapka Three Families of Functions Complexity One Lemma 1. <...> P There is a curious corollary from this lemma. <...> Consider a homogeneous linear equation with constant coefficients P(D)(z(x, y)) = 0 and let L be the space of its analytic solutions. <...> There is another case (case (4)) outside Lemma 1. <...> L-pairs A collection of functions forms a linear space if this collection is closed under addition and multiplication by a constant (complex numbers). <...> This means that a nonzero function of complexity 1 generates a linear space lying in Cl1. <...> In all cases it is not difficult to solve P Valery K.Beloshapka Three Families of Functions Complexity One Statement 4. <...> If r2 is not zero then from the second equation we have r(t) = ρ · emt + ˜ equation we have a(x) = α·emt+˜ are nonconstant functions. <...> The pseudo-group generated by the transformations (1), (2) and (3) we denote by G. The Valery K.Beloshapka Three Families of Functions Complexity One a1b1r1 (a1 −b1)2. <...> Each of them is a quadratic form in (c1, c2 <...>