By Vladislav V. Kravchenko
Pseudoanalytic functionality idea generalizes and preserves many the most important gains of advanced analytic functionality conception. The Cauchy-Riemann process is changed through a way more common first-order process with variable coefficients which seems to be heavily concerning very important equations of mathematical physics. This relation provides strong instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations through complex-analytic methods.
The publication is devoted to those contemporary advancements in pseudoanalytic functionality thought and their functions in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers attracted to complex-analytic tools, resolution recommendations for equations of mathematical physics, partial and traditional differential equations.
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Extra resources for Applied Pseudoanalytic Function Theory
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. 16) Theorem 33 (). 15).
Thus, we obtain the following connection between the stationary Schr¨ odinger equation and the system deﬁning p-analytic functions. Theorem 48. 37) where ν is a real-valued function and let W1 be another real-valued solution of this equation. 37). The following relation between solutions of the conductivity equation and p-analytic functions is valid also. Theorem 49. Let f be a positive continuously diﬀerentiable function in a domain Ω and let U be a real-valued solution of the equation div(f 2 ∇U ) = 0 2 in Ω.
40). Example 51. In the work  boundary value problems for p-analytic functions with p = x/(x2 + y 2 ) were studied. 1) with ν having the form 1 ν = − 2. 40). 1 Deﬁnition A generating sequence deﬁnes an inﬁnite sequence of Vekua equations. If for a given (original) Vekua equation we know not only a corresponding generating pair but the whole generating sequence, that is a pair of exact and independent solutions for each of the Vekua equations from the inﬁnite sequence of equations corresponding to the original one, we are able to construct an inﬁnite system of solutions of the original Vekua equation as is shown in the next deﬁnition.