Recent developments in applied pseudoanalytic function theory

Theorem 4 Let W be an (F, G)-pseudoanalytic function and let (F1, G1) be a successor of (F, G). Then · W is an (F1, G1)-pseudoanalytic function. This theorem shows us that to the difference of analytic functions whose derivatives are again analytic, the (F, G)-derivatives of pseudoanalytic func- tions are in general solutions of another Vekua equation. 6

Theorem 6 An (F, G)-derivative · W of an (F, G)-pseudoanalytic function W is (F, G)-integrable.

Theorem 7 Let (F, G) be a predecessor of (F1, G1). A continuous function is (F1, G1)-pseudoanalytic if and only if it is (F, G)-integrable.

Theorem 10 Let (F, G) be a generating pair in Ω. Let Ω1 be a bounded domain, Ω1 ⊂Ω. Then (F, G) can be embedded in a generating sequence in Ω1. Definition 11 A generating sequence {(Fm, Gm)} is said to have period µ > 0 if (Fm+µ, Gm+µ) is equivalent to (Fm, Gm) that is their characteris- tic coefficients coincide. Let W be an (F, G)-pseudoanalytic function. Using a generating sequence in which (F, G) is embedded we can define the higher derivatives of W by the recursion formula W [0] = W; W [m+1] =

Theorem 14 [8] The formal Taylor expansion (14) of a pseudoanalytic func- tion in formal powers defined by a periodic generating sequence converges in some neighborhood of the center. This theorem means only a local completeness of the system of formal powers. The following definition due to L. Bers describes the case when cor- responding formal powers represent a globally complete system of solutions of a Vekua equation much as in the case of usual powers of the variable z and the Cauchy-Riemann

Theorem 16 Let W be an (F, G)-pseudoanalytic function defined for |z −z0| < R. Then it admits a unique expansion of the form W(z) = P∞ n=0 Z(n)(an, z0; z) which converges normally for |z −z0| < θR, where θ is a positive constant depending on the generating sequence. The first version of this theorem was proved in [1]. We follow here [10].

Theorem 18 [10] A pseudoanalytic function defined in a simply connected domain can be expanded into a normally convergent series of formal polyno- mials (linear combinations of formal powers with positive exponents).

Theorem 20 [37] Let W be a pseudoanalytic function in a domain Ωbounded by a Jordan curve and satisfy the H¨older condition on ∂Ωwith the exponent α (0 < α ≤1). Then for any ε > 0 and any natural n there exists a pseudopolynomial of order n satisfying the inequality |W(z) −Pn(z)| ≤Const nα−ε for any z ∈Ω where the constant does not depend on n, but only on ε. 3 Solutions of second order elliptic equations as real components of complex pseudoana- lytic functions 3.1 Factorization of the stationar

Theorem 21 [30] Let f be a positive in Ωparticular solution of (16). Then for any real valued function ϕ ∈C2(Ω) the following equalities hold 1 4 (∆−ν) ϕ =  ∂z + fz f C   ∂z −fz f C  ϕ =  ∂z + fz f C

Proposition 22 [31] Let f be a positive particular solution of (16) and w be a solution of (19). Then the real valued function g = Sw is a solution of (16).

Proposition 23 [31] Let g be a real valued solution of (16). Then SPg = g + cf where c is an arbitrary real constant.

Theorem 21 together with Proposition 22 show us that equation (16) is equivalent to the Vekua equation (19) in the following sense. Every solution of one of these equations can be transformed into a solution of the other equation and vice versa. 15

Proposition 24 Let p and q be complex valued functions, p ∈C2(Ω) and p ̸= 0 in Ω. Then div p grad+q = p1/2(∆−r)p1/2 in Ω, (25) where r = ∆p1/2 p1/2 −q p.

Theorem 25 [32] Let p and q be real valued functions, p ∈C2(Ω) and p ̸= 0 in Ω, u0 be a positive particular solution of the equation (div p grad +q)u = 0 in Ω. (27) Then for any real valued continuously twice differentiable function ϕ the fol- lowing equality holds 1 4(div p grad+q)ϕ = p1/2  ∂z + fz f C   ∂z −fz f C

Theorem 29 [32] Let W = W1 + iW2 be a solution of (32). Then U = f −1W1 is a solution of the conductivity equation div(f 2∇U) = 0 in Ω, (34) and V = fW2 is a solution of the associated conductivity equation div(f −2∇V ) = 0 in Ω, (35) the function W1 is a solution of the stationary Schr¨odinger equation −∆W1 + r1W1 = 0 in Ω (36) with r1 = ∆f/f, and W2 is a solution of the associated stationary Schr¨odinger equation

Theorem 31 [32] Let W = W1 + iW2 be a solution of (32). Assume that f = p1/2u0, where u0 is a positive solution of (27) in Ω. Then u = p−1/2W1 is a solution of (27) in Ω, and v = p1/2W2 is a solution of the equation (div 1 p grad +q1)v = 0 in Ω, (41) where q1 = −1 p

Theorem 32 [30] Let W1 be a real valued solution of (36) in a simply con- nected domain Ω. Then the real valued function W2, solution of (37) such that W = W1 + iW2 is a solution of (32), is constructed according to the formula W2 = f −1A(if 2∂z(f −1W1)). (43) Given a solution W2 of (37), the corresponding solution W1 of (36) such that W = W1 + iW2 is a solution of (32), is constructed as follows W1 = −fA(if −2∂z(fW2)). (44) 20

Corollary 34 [32] Let U be a solution of (34). Then a solution V of (35) such that W = fU + if −1V is a solution of (32), is constructed according to the formula V = A(if 2Uz). (46) Conversely, given a solution V of (35), the corresponding solution U of (34) can be constructed as follows: U = −A(if −2Vz).

Corollary 35 [32] Let f = p1/2u0, where u0 is a positive solution of (27) in a simply connected domain Ωand u be a solution of (27). Then a solution 21

Proposition 36 Let W be a solution of (47). Then its (F, G)-derivative, the function w = · W is a solution of (19). 22

Proposition 37 Let w be a solution of (19). Then the function W(z) = Z z z0 w(ζ)d(F,G)ζ is a solution of (47). 3.5 p-analytic functions Definition 38 A function Φ = u + iv of a complex variable z = x + iy is said to be p-analytic in some domain Ωiff ux = 1 pvy, uy = −1 pvx in Ω

Theorem 39 Let f be a positive solution of the equation −∆g + νg = 0 (50) where ν is a real valued function and let W1 be another real valued solution of this equation. Then the function Φ = W1/f + ifW2, where W2 is defined by (43) is an f 2-analytic function, and vice versa, let Φ be an f 2-analytic function then the function W1 = f Re Φ is a solution of (50). The following relation between solutions of the conductivity equation and p-analytic functions is valid also.

Theorem 40 Let f be a positive continuously differentiable function in a domain Ωand let U be a real valued solution of the equation div(f 2∇U) = 0 in Ω. (51) Then the function Φ = U + iV is f 2-analytic in Ω, where V is defined by (46), and vice versa, let Φ be f 2-analytic in Ωthen U = Re Φ is a solution of (51). Thus, solutions of the stationary Schr¨odinger equation and of the con- ductivity equation can be converted into p-analytic functions and vice versa. In some cases this relation leads t

Theorem 44 Let u(z) be a solution of (27) defined for |z −z0| < R. Then it admits a unique expansion of the form u(z) = p−1/2(z) ∞ X n=0 Re Z(n)(an, z0; z) which converges normally for |z −z0| < R.

Theorem 45 An arbitrary solution of (27) defined in a simply connected domain where there exists a positive particular solution u0 can be expanded into a normally convergent series of formal polynomials multiplied by p−1/2.

Theorem 50 Let F = U(u)V (v) and G = i U(u)V (v) where U and V are arbitrary differentiable nonvanishing real valued functions, Φ = u + iv is an analytic function of the variable z = x + iy in Ωsuch that Φz is bounded and has no zeros in Ω. Then the generating pair (F, G) is embedded in the generating sequence (Fm, Gm), m = 0, ±1, ±2, . . .in Ωdefined as follows Fm = (Φz)m F and Gm = (Φz)m G for even m and Fm = (Φz)m U2 F

theorem 50 is bounded and has no zeros. References [1] Agmon S and Bers L 1952 The expansion theorem for pseudo-analytic functions. Proc. Amer. Math. Soc. 3 757-764. [2] Aleksandrov A Ya and Solovyev Yu I 1978 Spatial problems of elasticity theory: application of methods of the theory of functions of complex variable Moscow: Nauka (in Russian). [3] Astala K and P¨aiv¨arinta L 2006 Calder´on’s inverse conductivity problem in the plane. Annals of Mathematics, 163, No. 1, 265-299. [4] Begehr H 1985