By Vladislav V. Kravchenko
Pseudoanalytic functionality conception generalizes and preserves many the most important gains of advanced analytic functionality conception. The Cauchy-Riemann procedure is changed by means of a way more basic first-order approach with variable coefficients which seems to be heavily relating to vital equations of mathematical physics. This relation offers robust instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations through complex-analytic methods.
The ebook is devoted to those fresh advancements in pseudoanalytic functionality idea and their purposes in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers drawn to complex-analytic equipment, resolution suggestions for equations of mathematical physics, partial and usual differential equations.
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Extra resources for Applied Pseudoanalytic Function Theory
1/2 p p 24 Chapter 3. Second-order Equations Proof. , ). 8). The following statement is a generalization of Theorem 25. Theorem 29 (). Let p and q be real-valued functions, p ∈ C 2 (Ω) and p = 0 in Ω, u0 be a positive particular solution of the equation ( div p grad +q)u = 0 in Ω. 11) where f = p1/2 u0 . 12) Proof. 2). 12) is a solution of the equation (Δ − r)f = 0. 11). Remark 30. 13). 8) we have div p grad +q = p1/2 f −1 div f 2 grad f −1 p1/2 . 12) we obtain −1 2 div p grad +q = u−1 0 div pu0 grad u0 in Ω.
3. 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.
15) can be written as f ∂z (f −1 W1 ) + if −1 ∂z (f W2 ) = 0. 16) Theorem 33 (). 15). 20) where r2 = 2(∇f )2 /f 2 − r1 and (∇f )2 = fx2 + fy2 . Proof. 15) given in Remark 32. 17) from where we have that Re ∂z f 2 ∂z f −1 W1 where U = f −1 W1 . 16) by f −1 and applying ∂z gives 1 Δ f −1 W1 + i∂z f −2 ∂z (f W2 ) = 0 4 from where we have that Re ∂z f −2 ∂z (f W2 ) where V = f W2 . 18) 26 Chapter 3. 9) we have (Δ − r1 ) W1 = f −1 div(f 2 ∇ f −1 W1 ). 19). 20) for W2 it should be noticed that f div(f −2 ∇(f W2 )) = (Δ − r2 ) W2 .