By Vladislav V. Kravchenko
Pseudoanalytic functionality idea generalizes and preserves many an important positive factors of advanced analytic functionality thought. The Cauchy-Riemann approach is changed by way of a way more basic first-order approach with variable coefficients which seems to be heavily with regards to vital equations of mathematical physics. This relation provides robust instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations as a result of complex-analytic methods.
The ebook is devoted to those fresh advancements in pseudoanalytic functionality thought and their functions in addition to to multidimensional generalizations.
It is directed to undergraduates, graduate scholars and researchers drawn to complex-analytic equipment, resolution recommendations 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.
Proposition 40. 31). 4). function w = W This result can be veriﬁed also by a direct substitution. 33) z0 and we obtain the following statement. Proposition 41. 4). 31). 4) for solutions of the latter one, we obtain the following Cauchy integral theorem. Theorem 42. 4) in a domain Ω. Then for every closed curve Γ situated in a simply connected subdomain of Ω, Re Γ w dz + i Im f f wdz = 0. Γ Proof. By Theorem 18 w is (F, G)-integrable. That is Im Γ w dz − i Re f f wdz = 0. 5. 5 Cauchy’s integral theorem for the Schr¨ odinger equation Theorem 43 ( (Cauchy’s integral theorem for the Schr¨ odinger equation)).
The function s can be constructed in such a way that s(z0 ) = 0 at any ﬁxed point z0 ∈ Ω. Proof. The idea of the proof is simple. Given a pseudoanalytic function w, consider the functions w s=T a+b w and Φ = e−s w. 11) Apply ∂z to Φ: ∂z Φ = −∂z s · e−s w + e−s wz =− a+b w w e−s w + e−s (aw + bw) = 0. Thus, Φ is an analytic function in Ω. In order to make this reasoning rigorous we should prove that s is diﬀerentiable and consider the case when w has zeros in Ω. Assume that w is not identically zero (otherwise there is nothing to prove) and let Ω0 be the open subset of Ω in which w(z) = 0, z ∈ Ω0 .