By Vladislav V. Kravchenko
Pseudoanalytic functionality thought generalizes and preserves many the most important positive aspects of advanced analytic functionality thought. The Cauchy-Riemann method is changed via a way more common first-order method with variable coefficients which seems to be heavily concerning very important equations of mathematical physics. This relation offers strong instruments for learning and fixing Schrödinger, Dirac, Maxwell, Klein-Gordon and different equations through complex-analytic methods.
The e-book is devoted to those contemporary advancements in pseudoanalytic functionality conception and their purposes 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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26). 26) is uniquely determined up to an additive term cf −1 where c is an arbitrary real constant. 27) is proved in a similar way. Remark 37. 27) turn into the well-known formulas in complex analysis for constructing conjugate harmonic functions. Corollary 38. 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)).
5. 5 Cauchy’s integral theorem for the Schr¨ odinger equation Theorem 43 ( (Cauchy’s integral theorem for the Schr¨ odinger equation)). 1) in Ω. Then for every closed curve Γ situated in a simply connected subdomain of Ω, u f ∂z Re Γ f 2 ∂z dz + i Im Γ u f dz = 0. 35) Proof. 34) gives us the result. Remark 44. This theorem is also valid when f ≡ 1 that is for u being a harmonic function. 35) turns into the equality Γ ∂z udz = 0 which is obviously true because if u is harmonic, then ∂z u is analytic.
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. Remark 31. 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).
Applied Pseudoanalytic Function Theory by Vladislav V. Kravchenko