On a Class of A-Analytic Functions

Theorem 1.1 (see [3]). For any measurable on the complex plane C function A(z) : ∥A∥∞< 1 there exists unique homeomorphic solution χ(z) of the equation (1) which fixes the points 0, 1 and ∞. Note that if the function |A(z)| ⩽C < 1 is defined only in the domain D ⊂C, then it can be extended to the whole C by setting A ≡0 outside D, so the Theorem 1.1 holds for any domain D ⊂C. ∗sadullaev@mail.ru †jabborov61@mail.ru c⃝Siberian Federal University. All rights reserved – 374 –

Theorem 1.2 (see [5,6]). The set of all generalized solutions of equation (1) is exhausted by the formula f(z) = Φ[χ(z)], where χ(z) is a homeomorphic solution from Theorem 1.1, and Φ(ξ) is a holomorphic function in the domain χ(D). Moreover, if the generalized solution f(z) has isolated singular points, the holomorphic function Φ = f ◦χ−1 also has isolated singular points of the same types. From Theorem 1.2 implies that the A-analytic function f carries out internal mapping, i.e. it mapping an

Theorem 1.3 (see [8]). If a function A(z) belongs to the class of m-smooth functions (A(z) ∈ Cm(D)), then every solution f of the equation (1) also belongs to, at least, the class, i.e. f ∈ Cm(D). The purpose of this paper is to study A-analytic functions in a particular case, when the function A(z) is anti-holomorphic in a considered domain. As we can see below, in this special case the solution of (1) possesses many properties of analytic functions, has an excellent integral representation, an

Theorem 2.1. (analogue of Cauchy’s theorem [16]). If f ∈OA(D) ∩C( ¯D), where D ⊂C is a domain with rectifiable boundary ∂D, then ∫ ∂D f(z)(dz + A(z)d¯z) = 0. – 375 –

Theorem 2.2. K(z, ξ) is A-analytic function outside of the point z = ξ, i.e. K ∈OA(D \ {ξ}). Moreover, at z = ξ the function K(z, ξ) has a simple pole.

Theorem 2.4. A differential of the current is coincide with the Dirac measure δξ, i.e. for any finite infinitely smooth in D function φ ∈F 0(D) we have dω ◦φ = ∫ ω ∧dφ = φ(ξ), α ∈F 0(D).

Theorem 2.5 (Cauchy formula [18]). Let D ⊂C is an arbitrary convex domain and G ⊂D is a subdomain, with piecewise smooth boundary ∂G. Then for any function f(z) ∈OA(G) ∩C( ¯G) we have a formula f(z) = ∫ ∂G K(ξ, z)f(ξ)(dξ + A(ξ)d¯ξ), z ∈G. (3) – 377 –

Theorem 3.1. If f(z) ∈OA(L(a, r)) ∩C(¯L(a, r)), where L(a, r) = {ξ ∈D : |ψ(ξ, a)| < r} ⊂⊂D is a lemniscate, then the function f(z) can be expanded to the Taylor series in L(a, r): f(z) = ∞ ∑ k=0 cjψk(z, a), (5) where ck = 1 2πi ∫ ∂L(a, ρ)

Theorem 3.2 (Laurent expansion). Let f(z) be A-analytic in a ring of lemniscates: f ∈ OA(L(a, R) \ L(a, r)), r < R. Then f(z) will be expanded to the Loran series in this ring: f(z) = ∞ ∑ k=−∞ cjψk(z, a), (6) where a coefficients of Taylor-Laurent series which determines by the formula ck = 1 2πi ∫ ∂L(a, ρ) f(ξ)

Theorem 4.1. If a is an essential singular point of the A-analytic function f(z), then f does not accept more than two (exceptional) points of extended complex plane C. In the proof of the theorem, by analogy of analytic functions, essentially used Montel’s theorem on compact (normal) family of A-analytic functions and modular analytic function in the unit disk U = U(0, 1). Recall that the family of functions {fα(z)}α∈Λ ⊂OA(G), is called normal if every its subfamily {fα(z)}α∈Λ0, Λ0 ⊂Λ contains

Theorem 4.2 (Montel). A locally uniformly bounded family of A-analytic functions {fα(z)}α∈Λ ⊂OA(G) forms a normal family. To proof this theorem using Arzela’s theorem it is enough to show equicontinuity of the family {fα(z)}α∈Λ on any compact K ⊂⊂G, that is ∀ε > 0 ∃δ > 0 : ∀z′′, z′ ∈K, |z′′ −z′| < δ, ∀f ∈ {fα} ⇒|f(z′′) −f(z′)| < ε. – 380 –

Corollary 4.3. If each function of family of A-analytic functions {fα(z)}α∈Λ do not accept two values a ∈C, b ∈C, a ̸= b, then this family is normal. In fact, considering the family {fα (z) −a b −a } α∈Λ , we can assume that a = 0, b = 1. We use the modular function µ (z) : U →C \ {0, 1}. Note that the modular function µ(z) conform maps the circular triangle ∆0 = ABC ⊂¯U, A, B, C ∈∂U, with sides AB⊥∂U, BC⊥∂U, CA⊥∂U on the upper half of the plane so that µ(A) = 0, µ(B) = 1, µ(C) = ∞. Then, it ext