By Vladislav V. Kravchenko

Pseudoanalytic functionality idea generalizes and preserves many the most important positive factors of advanced analytic functionality thought. The Cauchy-Riemann method is changed via a way more common first-order procedure with variable coefficients which seems to be heavily with regards to vital equations of mathematical physics. This relation provides strong instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations because of complex-analytic methods.

The publication is devoted to those fresh 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, answer suggestions for equations of mathematical physics, partial and traditional differential equations.

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**Extra resources for Applied Pseudoanalytic Function Theory **

**Example text**

Here we follow the deﬁnitions and notations introduced by L. Bers due to their complete structural resemblance with the classical 56 Chapter 5. Cauchy’s Integral Formula results from analytic function theory. 1) in a simply connected, bounded domain Ω where a and b are complex-valued functions satisfying the H¨ older condition up to the boundary. L. 2) where α is any complex number. 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.

37) where ν is a real-valued function and let W1 be another real-valued solution of this equation. 37). The following relation between solutions of the conductivity equation and p-analytic functions is valid also. Theorem 49. Let f be a positive continuously diﬀerentiable function in a domain Ω and let U be a real-valued solution of the equation div(f 2 ∇U ) = 0 2 in Ω. 38). 6. p-analytic functions 33 Thus, solutions of the stationary Schr¨ odinger equation and of the conductivity equation can be converted into p-analytic functions and vice versa.

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 [1]. We follow here [15]. Remark 66. Necessary and suﬃcient conditions for the relation θ = 1 are, unfortunately, not known. However, in [15] the following suﬃcient conditions for the case when the generators (F, G) possess partial derivatives are given. 5. The Runge theorem 43 for some ε > 0.