By Vladislav V. Kravchenko

This ebook covers contemporary advancements in pseudoanalytic functionality thought together with purposes and multidimensional generalizations. Written at an undergraduate point, it's the first publication of the idea utilized to many types of mathematical physics.

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4) in the following sense. Every solution of one of these equations can be transformed into a solution of the other equation and vice versa. 2 Factorization of the operator div p grad +q. , [96]). Here we formulate it as an operational relation. Proposition 28. Let p and q be real-valued functions, p ∈ C 2 (Ω) and p = 0 in Ω. 8) div p grad +q = p1/2 (Δ − r)p1/2 where r= Δp1/2 q − . 1/2 p p 24 Chapter 3. Second-order Equations Proof. , [118]). 8). The following statement is a generalization of Theorem 25.

10). In order to obtain the second assertion of the theorem, let us show that p1/2 div 1 grad +q1 (p1/2 ϕ) = f div(f −2 ∇(f ϕ)) p for any real-valued ϕ ∈ C 2 (Ω). 9), f div(f −2 ∇(f ϕ)) = Δ− Δf −1 f −1 ϕ = (Δ − r2 ) ϕ. Straightforward calculation gives us the equality Δf −1 3 = −1 f 4 ∇p p 2 − 1 Δp + 2 p ∇p ∇u0 , p u0 − Δu0 +2 u0 ∇u0 u0 2 . 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 Δp−1/2 Δf −1 = −1/2 + 2 −1 f p ∇p p 2 ∇p ∇u0 , p u0 − q +2 p ∇u0 u0 2 .

1 Preliminary information on the Cauchy integral formula for pseudoanalytic functions The subject of Cauchy integral formulas for classes of pseudoanalytic functions has been treated by many diﬀerent mathematical schools and with implementation of diﬀerent notations and techniques. 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.