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Some Extensive Discussions of Liouville's Theorem and Cauchy's Integral Theorem on Structural Holomorphic

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Some Extensive Discussions of Liouville's Theorem and Cauchy's Integral Theorem on Structural Holomorphic Some extensive discussions of Liouville’s theorem and Cauchy’s integral theorem on structural holomorphic Gen Wang ∗ Department of Mathematics, Zhejiang Normal University, ZheJiang, Jinhua 321004, China. Abstract Classic complex analysis is built on structural function K = 1 only associated with Cauchy-Riemann equations, subsequently various generalizations of Cauchy-Riemann equations start to break this situation. The goal of this article is to show that only structural function K = Const such that Liouville’s theorem is held, otherwise, it’s not valid any more on complex domain based on structural holomorphic, the correction K should be w =Φe− , whereΦ = Const. Those theories in complex analysis which keep constant are unable to be held as constant in the framework of structural holomorphic. Synchronously, it deals with the generalization of Cauchy’s integral theorem by using the new perspective of structural holomorphic, it is also shown that some of theories in the complex analysis are special cases at K = Const, which are narrow to be applied such as maximum modulus principle. Contents 1 Introduction 2 1.1 Structuralholomorphic.............................. 2 1.2 Liouville’s theorem . 6 1.3 Maximum modulus principle . 7 1.4 Cauchy’sintegraltheorem . .. .. 8 2 Generalized Cauchy’s integral theorem 10 arXiv:1807.02600v2 [math.CV] 24 Feb 2020 3 Liouville’s theorem on structural holomorphic 11 3.1 Maximum modulus principle on structural holomorphic . 12 ∗[email protected] 1 G. Wang 1 Introduction As all known, Liouville’s theorem and maximum modulus principle all describe the con- stant properties on different conditions, Cauchy’s integral theorem also can be generalized as well [2, 5, 6, 11]. On structural holomorphic [1] of them might be modified to suit general situation. 1.1 Structural holomorphic In the field of complex analysis, the Cauchy-Riemann equations as a certain continuity and differentiability criteria for a complex function w to be complex differentiable, namely, holomorphic or analytic [1, 3]. ∂ ∂ u ∂w ∂x − ∂y =0 =0 (1) ∂ ∂ v ⇔ ∂z ∂y ∂x Subsequently, the Carleman-Bers-Vekua (CBV) equation follows [7–10] ∂w + Aw + Bw =0 ∂z where A = 1 a + d + √ 1c √ 1b , B = 1 a d + √ 1c + √ 1b , or equivalently in 4 − − − 4 − − − a matrix form u O =0 (2) v ∂ + a ∂ b b where O = ∂x ∂y is operator matrix. ∂ − ∂ − ∂y + c ∂x + d ! Furthermore,b the boundary value problem was generally investigated: wz + A (z) w + B (z) w = H, z < 1 k | | Re z− w (z) = g (z) , z =1 | | What’s more, the nonlinear Cauchy-Riemann (NCR) equations is proposed [4] u = v + f (u, v) ,u = v + g (u, v) y − x x y The matrix form is ∂ ∂ u g (u, v) ∂x − ∂y = ∂ ∂ v f (u, v) ∂y ∂x where f(u, v) and g(u, v) are given functions. the system of inhomogeneous Cauchy-Riemann equations is ∂u ∂v ∂u ∂v = G(x, y), + = F (x, y) ∂x − ∂y ∂y ∂x for some given functions G(x, y) and F (x, y) defined in an open subset of R2. Structural holomorphic can be expressed in the form a u = a v + f (u, v) , a u = a v + g (u, v) 1 y − 2 x 3 x 4 y 2 G. Wang where ai, i =1, 2, 3, 4 are real function. the structural holomorphic can be rewritten ∂w ∂κ + w =0 ∂z¯ ∂z¯ Above all, one has retrospected the developing history of relating holomorphic conditions for a complex function w defined on the complex domain C. In conclusion, the evolution of Cauchy-Riemann equations can be shown in a chain ∂w ∂w ∂w =0 = ϕ (z, z) + Aw =0 ∂z¯ → ∂z¯ → ∂z¯ ∂w ∂w ∂K ∂w ∂K + Aw + Bw =0 K + w =0 + w =0 → ∂z¯ → ∂z¯ ∂z¯ → ∂z¯ ∂z¯ ∂w ∂w ∂K Note that it’s a trivial term K ∂z¯ for this K in the K ∂z¯ + w ∂z¯ = 0, then we only study ∂w ∂K structural holomorphic ∂z¯ + w ∂z¯ = 0. And more general formally equation can be ∂w ∂K + w = ϕ (z, z) ∂z¯ ∂z¯ where K is structural function which can be chosen arbitrarily, ϕ (z, z) is a given function, it becomes the structural holomorphic as ϕ (z, z) = 0. Eventually, most of equations for holomorphic condition are not specified, especially, the coefficients in the equation are indistinct, particularly, the theories for the generalization of Cauchy-Riemann equation most of them come from other branches such as partial differen- tial equation, or physical problems which are not from the pure complex analysis. Surely, the generalizations of Cauchy-Riemann equation also get new difficulties and complexity emerged. On account of this point, [1] has restarted with the continuity and differentiability of a complex function w as usually done to discuss in a K-transformation method given by w (z) w (z)= w (z) K (z) → where structural function K (z) = k + √ 1k is complex valued function on Ω, k ,k are 1 e 2 1 2 real functions with respect to the variables− x, y. Mathematically, this K-transformation is the most natural way to generalize the Cauchy-Riemann equation in complex analysis. As traditionally depicted as K = 1, or the parts take values k1 =1,k2 = 0. Mathemati- cally, the generalized holomorphic condition can be clearly expressed by structural function K (z), and all equations for developing the Cauchy-Riemann equation can be summarized as some of inevitable connections to the structural function K (z) here. Therefore it follows the main theorem given by Theorem 1. [1] Let Ω Cn be an open set and complex valued function w (z)= u + √ 1v is said to be a structural⊂ holomorphic on Ω if and only if − D ∂w ∂K w = + w =0, i =1, 2, , n (3) ∂zi ∂zi ∂zi ··· and its solutions are called structural analytic functions. 3 G. Wang As pictured in [1], structural holomorphic differential equation can unify all situations on the way to develop and generalize the Cauchy-Riemann equation in a simple and compact differential form, since K (z) can be chosen arbitrarily. Thusly, it suits either functions of one complex variable z in C or functions of several complex variables z =(z1, z2, , zn)T Cn. Hence, if one consider (3) in Ω C1 = C, then (3) is equivalent to the following··· matrix∈ equation ⊂ k k ∂ ∂ v ∂ ∂ k k u 1 2 ∂x ∂y = ∂x ∂y 2 1 k k ∂ ∂ u ∂ ∂ −k −k v − 2 1 − ∂y ∂x − ∂y ∂x − 1 2 k k ∂ ∂ ∂ ∂ k k Notice that here 1 2 ∂x ∂y and ∂x ∂y 2 1 are structural matrix k k ∂ ∂ ∂ ∂ −k −k − 2 1 − ∂y ∂x − ∂y ∂x − 1 2 operators only associated with the Wirtinger derivatives and structural function K (z). As a result, since K (z) can be arbitrarily chosen, then k1 and k2 are chosen arbitrarily. To assume that K (z)=1+ κ (z) in C which is prominently showing the classic complex analysis built on 1. The structural holomorphic is ∂w ∂κ = w ∂z − ∂z Conversely, it laterally reveals the importance of structural function to enlarge the scope of application. (3) can be written in a more general form D ∂w ∂K w = + w = ϕ (z, z) ∂zi ∂zi ∂zi where ϕ (z, z) is some given function. Here one gives a classical result below as a reference Lemma 1 ( [2,6]). If w : Ω C is analytic, then w preserves angles at each point z of Ω, → 0 where w′ (z ) =0. 0 6 This lemma will tell us that w preserves angles at each point z0 of Ω in the structural holomorphic. Accordingly, a natural corollary can be deduced in Cn, Corollary 1. The structural holomorphic for w (z)= u + √ 1v on Ω Cn is − ⊂ Dw = ∂w + w∂K =0 where K (z) is structural function defined on Ω. If K = c C takes values as complex constant, then c∂w = 0 always holds, in particular, let K = 1 be∈ given for ∂w = 0. In the same way, K (z)=1+ κ (z) holds for ∂w = w∂κ − It strongly implies the necessary of the structural function κ (z) for any generalization of Cauchy-Riemann equation. 4 G. Wang Let all structural holomorphic function be a collection, denoted as Shol. Note that corol- lary 1 is complete and compact form including all possible structural holomorphic condition because of K (z) can be chosen arbitrarily, say, given a point z Ω Cn, then Dw = 0, 0 ∈ ⊂ |z=z0 that is ∂w + w (z ) ∂K =0 |z=z0 0 |z=z0 Suppose that w (z0) = 0, then the equation ∂w =0 |z=z0 holds at z0, accordingly, there is ∂w z=z0 = 0 reduced to the classic Cauchy-Riemann equa- tion. | C D ∂w ∂K To consider the z0 Ω , then (3) becomes ∂z w = ∂z + w ∂z = 0 as n = 1 with D ∂ ∂K ∈ ∂⊂ ∂z w = ∂z w + w ∂z , hence ∂z w = 0 holds at z0 in Ω, according to the lemma 1, structural holomorphic always preserves angles6 except some particular situations, A natural question arises while investigating the above-mentioned differential systems structural holomorphic condition: how should the solutions be obtained. In fact, the solution of (3) in C or Cn is a little hard to get owing to the specific expression of K (z) is unknown, but for some particular circumstances, one can obtain the formal solution. For a reason that K (z) can be arbitrarily chosen, for instance, taking the form of K is n n 1 n 2 2 K (z)= z + a1z − + a2z − + + an 2z + an 1z + an ··· − − then structural holomorphic condition is ∂w n 1 n 2 n 3 = w nz − + a1 (n 1) z − + a2 (n 2) z − + +2an 2z + an 1 ∂z − − − ··· − − where ai C, or ai R, i = 1, 2, , n, hence choose origin z = 0 such that structural ∈ ∈ ∂w ··· holomorphic condition turns to + wan 1 = 0.
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