By Vladislav V. Kravchenko

Pseudoanalytic functionality concept generalizes and preserves many an important positive aspects of complicated analytic functionality concept. The Cauchy-Riemann procedure is changed through a way more normal first-order approach with variable coefficients which seems to be heavily on the topic of very important equations of mathematical physics. This relation provides strong instruments for learning and fixing Schr?dinger, Dirac, Maxwell, Klein-Gordon and different equations due to complex-analytic tools. The booklet is devoted to those fresh advancements in pseudoanalytic functionality conception and their functions in addition to to multidimensional generalizations. it truly is directed to undergraduates, graduate scholars and researchers drawn to complex-analytic tools, resolution concepts for equations of mathematical physics, partial and traditional differential equations.

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Theorem 57 (Removable singularity theorem). Let w be pseudoanalytic and bounded for 0 < |z − z0 | < r for some number r > 0. Then w can be deﬁned at z0 in such a way that it is pseudoanalytic in the whole disk |z − z0 | < r. Proof. 9) is unchanged if a point is removed from its region of integration. Theorem 58 (Similarity principle). Let w be pseudoanalytic in a domain Ω. 10) 40 Chapter 4. Formal Powers and vice versa, let Φ be an analytic function deﬁned in a domain Ω. 10) is pseudoanalytic in Ω.

By Theorem 58, w = Φes where Φ is an entire function and s is bounded (due to the fact that a and b have compact support). Thus we have that Φ is bounded and hence, by Liouville’s theorem in analytic function theory, is a constant. Since w(z0 ) = 0, this constant must be zero, and the corollary follows. 4. Taylor series in formal powers 41 Corollary 61 (Maximum modulus theorem). Let w be a pseudoanalytic function deﬁned in Ω and continuous in Ω. Then there exists a positive constant M ≥ 1 depending only on a and b such that, for z in Ω, |w(z)| ≤ M max |w(t)| .

28): ψ = A(if 2 φz ). It can be veriﬁed that the expression A(if 2 φz ) makes sense, that is ∂x (f 2 φx ) + ∂y (f 2 φy ) = 0. 20). 26). 26) is uniquely determined up to an additive term cf −1 where c is an arbitrary real constant. 27) is proved in a similar way. Remark 37. 27) turn into the well-known formulas in complex analysis for constructing conjugate harmonic functions. Corollary 38. 17). 15), is constructed according to the formula V = A(if 2 Uz ). 4. 17) can be constructed as U = −A(if −2 Vz ).