By Vladislav V. Kravchenko
Pseudoanalytic functionality concept generalizes and preserves many an important positive factors of complicated analytic functionality idea. The Cauchy-Riemann method is changed through a way more basic first-order method with variable coefficients which seems to be heavily with regards to vital equations of mathematical physics. This relation offers strong instruments for learning and fixing Schr?dinger, Dirac, Maxwell, Klein-Gordon and different equations as a result of complex-analytic equipment. The booklet is devoted to those contemporary advancements in pseudoanalytic functionality conception and their functions in addition to to multidimensional generalizations. it's directed to undergraduates, graduate scholars and researchers attracted to complex-analytic equipment, answer strategies for equations of mathematical physics, partial and traditional differential equations.
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Extra resources for Applied Pseudoanalytic Function Theory (Frontiers in Mathematics)
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 ). Proof. 27). Corollary 39. 10). 15), is constructed according to the formula −1 2 v = u−1 0 A(ipu0 ∂z (u0 u)). 15), is constructed according to the formula u = −u0 A(ip−1 u−2 0 ∂z (u0 v)). Proof. 27). 15): Wz = fz W. 30). As was pointed out in Remark 34, the pair of functions: F = f and G = fi is a generating pair for this equation.
Assume that w is not identically zero (otherwise there is nothing to prove) and let Ω0 be the open subset of Ω in which w(z) = 0, z ∈ Ω0 . 11) makes sense and Φ is analytic in Ω0 . The analyticity of Φ in Ω\Ω0 is proved with the aid of Theorem 57. We omit this part of the proof, referring the reader to [30, p. 142] or [32, Chapter 4]. The second part of the theorem is based mainly on the same ideas and can also be found in [30, p. 142] or [32, Chapter 4]. From the similarity principle a variety of function theoretic results for pseudoanalytic functions can be deduced.
This follows from Theorem 58 and the maximum modulus principle for analytic functions. 12) n=0 where the absence of the subindex m means that all the formal powers correspond to the same generating pair (F, G) and the series converges uniformly in some neighborhood of z0 . Based on the similarity principle it can be shown  that the uniform limit of pseudoanalytic functions is pseudoanalytic, and that a uniformly convergent series of (F, G)-pseudoanalytic functions can be (F, G)-diﬀerentiated term by term.