By Vladislav V. Kravchenko
Pseudoanalytic functionality concept generalizes and preserves many an important good points of advanced analytic functionality conception. The Cauchy-Riemann approach is changed by means of a way more normal first-order approach with variable coefficients which seems to be heavily relating to very important equations of mathematical physics. This relation provides robust instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations using complex-analytic methods.
The e-book is devoted to those contemporary advancements in pseudoanalytic functionality conception and their functions in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers attracted to complex-analytic tools, answer suggestions for equations of mathematical physics, partial and usual differential equations.
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Extra info for Applied Pseudoanalytic Function Theory
The maximal error maxz∈Ω |u(z) − u(z)| where u is the exact solution and u = 21 n=1 an un , the real constants an being found by the collocation method, was of order 10−7 . A very fast convergence of the method was observed. Although the numerical method based on the usage of explicitly or numerically constructed pseudoanalytic formal powers still needs a much more detailed analysis, these results show us that it is quite possible that in due time it can rank high among other numerical approaches.
18) was observed in  and turned out to be essential for solving the Calder´on problem in the plane. Theorem 35 (). 15). 10) in Ω. 24) 2 . 3. Conjugate metaharmonic functions 27 Proof. 17). 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 .
The transplant operator 59 where η = 2(∇f )2 /f 2 − ν. 7) can be constructed explicitly: W2 = f −1 A(if 2 ∂z (f −1 W1 )). 9) and consider the corresponding Vekua equation gz wz = w in Ω. 9), meanwhile Im W and Im w satisfy the following, in general diﬀerent Schr¨ odinger equations, in Ω (−Δ + η1 ) Im W = 0 and (−Δ + η2 ) Im w = 0 in Ω where η1 = 2(∇f )2 /f 2 − ν and η2 = 2(∇g)2 /g 2 − ν. 12) acting in the following way: Tf,g [W ] = P + W + ig −1 A(ig 2 ∂z (g −1 P + W )). 12). This is why we call the operator Tf,g the transplant operator.
Applied Pseudoanalytic Function Theory by Vladislav V. Kravchenko