By Siemons J.
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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).
2-Designs and a differential equation by Siemons J.