By Vladislav V. Kravchenko
Pseudoanalytic functionality concept generalizes and preserves many an important gains of complicated analytic functionality conception. The Cauchy-Riemann process is changed via a way more common first-order procedure with variable coefficients which seems to be heavily regarding very important equations of mathematical physics. This relation provides strong instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations as a result of complex-analytic methods.
The booklet is devoted to those contemporary advancements in pseudoanalytic functionality idea and their functions in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers drawn to complex-analytic tools, resolution recommendations for equations of mathematical physics, partial and usual differential equations.
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Pseudoanalytic functionality concept generalizes and preserves many the most important beneficial properties of complicated analytic functionality thought. The Cauchy-Riemann method is changed through a way more basic first-order procedure with variable coefficients which seems to be heavily relating to very important equations of mathematical physics.
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Extra resources for Applied Pseudoanalytic Function Theory
Let W be an (F, G)-pseudoanalytic function deﬁned for |z − z0 | < R. (n) (an , z0 ; z) which Then it admits a unique expansion of the form W (z) = ∞ n=0 Z 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 . We follow here . Remark 66. Necessary and suﬃcient conditions for the relation θ = 1 are, unfortunately, not known. However, in  the following suﬃcient conditions for the case when the generators (F, G) possess partial derivatives are given.
This function is denoted as w(z) = Z (−1) (α, z0 , z). Note that, while no other condition is imposed, this function is not unique. The following generalization of the Cauchy integral formula is valid. Theorem 81 (). 1) deﬁned in a domain Ω bounded by a simple closed continuously diﬀerentiable curve Γ and assume that W is continuous up to the boundary Γ. Then for any z ∈ Ω the following equality holds: W (z) = 1 2π Z (−1) (iW (ζ)dζ, ζ, z). 3) Γ This integral should be understood in the following sense.
Second-order Equations Proof. Consider ∂z + fz C f 2 ∂z − ∂z f 1 |∂z f | fz C ϕ = Δϕ − ϕ − ∂z ϕ f 4 f2 f 1 1 Δf ϕ = (Δ − ν) ϕ. 2). 2). The operator ∂z − in the form fz f I, where I is the identity operator, can be represented ∂z − fz I = f ∂z f −1 I. f Let us introduce the notation P = f ∂z f −1 I. 1) into solutions of the Vekua equation ∂z + fz C w = 0. 17). Then it is clear that for such w we have that P Sw = w. Proposition 26. 4). 1). Proof. 17). Let u = Re w and v = Im w. Consider ∂y Φ1 + ∂x Φ2 = 1 f (∂y u + ∂x v) − ∂x f ∂y f u+ v f f .
Applied Pseudoanalytic Function Theory by Vladislav V. Kravchenko