By Vladislav V. Kravchenko
Pseudoanalytic functionality conception generalizes and preserves many an important positive factors of complicated analytic functionality thought. The Cauchy-Riemann method is changed via a way more normal first-order approach with variable coefficients which seems to be heavily regarding vital equations of mathematical physics. This relation offers robust instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations by using complex-analytic methods.
The ebook is devoted to those contemporary advancements in pseudoanalytic functionality conception and their purposes in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers attracted to complex-analytic tools, answer strategies for equations of mathematical physics, partial and traditional differential equations.
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Extra resources for Applied Pseudoanalytic Function Theory
48 Chapter 4. Formal Powers The following step is to construct the generating pair (F2 , G2 ). 4) and apply the (F1 , G1 )-derivative to them. 4) written in the form ϕz F1 + ψz G1 = 0 which in our case can be represented as ϕz V U + ψz i = 0. 21) Again, let us look for a solution in the form ϕ = ϕ(u), ψ = ψ(v). 21) is satisﬁed if ϕ (u) = U 2 (u)du ϕ(u) = 2 ψ (v) from which we obtain and ψ(v) = V 2 (v)dv. 4) has the form U 2 (u)duΦz w1 = V (v) + U (u) V 2 (v)dvΦz iU (u) . V (v) Its (F1 , G1 )-derivative is obtained as w˙ 1 = Φz U V uz + iΦz U V vz = U V (Φz )2 .
4) Thus, in order to be able to write down Z (−1) (α, z0 , z) for any complex coeﬃcient α, which is required for the Cauchy integral formula, we need to construct two Cauchy kernels: Z (−1) (1, z0 , z) and Z (−1) (i, z0 , z). 2. 3) possesses many important properties similar to those of a usual Cauchy integral including Plemelj-Sokhotski formulas and others, which are indispensable for solving corresponding boundary value problems. 2 Relation between the main Vekua equation and the system describing p-analytic functions Let us consider the case when a ≡ 0 and b = fz /f where f is a continuously diﬀerentiable real-valued positive function deﬁned in Ω, that is the main Vekua equation.
28) ψz − if 2 φz = 0 is valid. Note that if W1 = Re W , then φ = W1 /f . 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 ).