By Vladislav V. Kravchenko

Pseudoanalytic functionality conception generalizes and preserves many an important positive aspects of complicated analytic functionality thought. The Cauchy-Riemann process is changed by way of a way more basic first-order procedure with variable coefficients which seems to be heavily concerning vital equations of mathematical physics. This relation offers strong instruments for learning and fixing Schr?dinger, Dirac, Maxwell, Klein-Gordon and different equations via complex-analytic equipment. The e-book is devoted to those contemporary advancements in pseudoanalytic functionality thought and their functions in addition to to multidimensional generalizations. it's directed to undergraduates, graduate scholars and researchers drawn to complex-analytic tools, resolution options for equations of mathematical physics, partial and traditional differential equations.

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50 Chapter 4. Formal Powers Consider Φz Φz b(F,G) − 2uz U U U uz + ivz U V uz + vz − 2uz uz − ivz U V U V U U V uz iuz = uz − +i − uz + = uz U V U V = . 22) is proved in the case of m being odd. Now let m be even. 22) is valid iﬀ the expression We have Φz uz + ivz b(F,G) = uz − ivz Φz m = Φz b Φz (F,G) uz uz U uz . U is equal to −B(F,G) + 2 UU uz . U V uz + iuz U V = uz V U +i U V and from the other side −B(F,G) + 2 U uz = −uz U V U −i U V +2 U uz = uz U V U +i U V . 22) is proved in all cases and the sequence (Fm , Gm ), m = 0, ±1, ±2, .

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. Then the corresponding characteristic coeﬃcients A(F,G) and B(F,G) have the form A(F,G) = 0, B(F,G) = fz , f 30 Chapter 3.

Let q ≡ 0. Then u0 can be chosen as u0 ≡ 1. 11) gives us the equality 1 ∂z p1/2 div(p grad ϕ) = p1/2 ∂z + 1/2 C 4 p ∂z − ∂z p1/2 C (p1/2 ϕ). 2. Factorization of the operator div p grad +q. 10) which we denote by u0 . Let f be a real function of x and y. Consider the Vekua equation Wz = fz W f in Ω. 15) This equation plays a crucial role in all that follows, hence we will call it the main Vekua equation. 2). Denote W1 = Re W and W2 = Im W . Remark 32. 15) can be written as f ∂z (f −1 W1 ) + if −1 ∂z (f W2 ) = 0.