By Vladislav V. Kravchenko

Pseudoanalytic functionality thought generalizes and preserves many an important positive factors of advanced analytic functionality thought. The Cauchy-Riemann process is changed through a way more basic first-order process with variable coefficients which seems to be heavily concerning vital equations of mathematical physics. This relation offers robust instruments for learning and fixing Schr?dinger, Dirac, Maxwell, Klein-Gordon and different equations using complex-analytic equipment. 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 equipment, answer concepts for equations of mathematical physics, partial and traditional differential equations.

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Theorem 21. If W is a continuous function deﬁned in a simply connected domain Ω, and if W is (F, G)-integrable, then there exists an (F, G)-pseudoanalytic function w in Ω, such that W (z) = d(F,G) w(z) . dz Proof. Under the hypotheses of the theorem the function ω = ϕ + iψ = ∗ W d(F,G) z Γ is well deﬁned and possesses continuous partial derivatives (z0 being any ﬁxed point in Ω). 3. Derivatives and integrals of pseudoanalytic functions 19 from where it follows that ϕz = GW , FG − FG ψz = − FW FG − FG and ϕz = − GW , FG − FG ψz = FW .

10) we obtain the equality − Δu0 q = + u0 p ∇p ∇u0 , p u0 . 1 Δp +2 2 p ∇p ∇u0 , p u0 + Thus, Δf −1 3 = f −1 4 Notice that Then ∇p p 2 − Δp−1/2 3 = 4 p−1/2 Δf −1 Δp−1/2 = +2 f −1 p−1/2 ∇p p 2 ∇p ∇u0 , p u0 − q +2 p ∇u0 u0 2 . 1 Δp . 2 p + q +2 p ∇u0 u0 2 . 8). 15) are solutions of associated stationary Schr¨ odinger equations, being also related to conductivity equations as 28 Chapter 3. 24). The following natural question arises then. We know that given an arbitrary real-valued harmonic function in a simply connected domain, a conjugate harmonic function can be constructed explicitly such that the obtained pair of harmonic functions represent the real and imaginary parts of a complex analytic function.

The meaning of this result for the stationary Schr¨ odinger equation and for the conductivity equation we discuss separately in the next two subsections. 23) where ν is a real-valued function. 19). Example 79. 23) depend on one Cartesian variable: ν = ν(x). Suppose we are given a particular solution f = f (x) of the ordinary diﬀerential equation − d2 f + νf = 0. 24) This solution is suﬃcient for the application of our result from the preceding section for constructing the corresponding generating sequence and hence the system of formal powers for the main Vekua equation which in this particular case has the form fx Wz = W.