Piecewise Weierstrass preparation and division for o-minimal holomorphic functions

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

Coefficients problems for families of holomorphic functions related to hyperbola

Mathematica Slovaca ◽

10.1515/ms-2017-0375 ◽

2020 ◽

Vol 70 (3) ◽

pp. 605-616

Author(s):

Stanisława Kanas ◽

Vali Soltani Masih ◽

Ali Ebadian

Keyword(s):

Unit Disk ◽

Holomorphic Functions ◽

Hankel Determinant ◽

Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

Demonstratio Mathematica ◽

10.1515/dema-2019-0043 ◽

2019 ◽

Vol 52 (1) ◽

pp. 482-489 ◽

Cited By ~ 1

Author(s):

Andriy Bandura ◽

Oleh Skaskiv ◽

Liana Smolovyk

Keyword(s):

Partial Differential Equations ◽

Continuous Function ◽

Differential Equations ◽

Sufficient Conditions ◽

Holomorphic Functions ◽

Positive Continuous Function ◽

Partial Differential ◽

Partial Derivatives ◽

Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000045 ◽

2021 ◽

pp. 1-29

Author(s):

ALEXANDER BRUDNYI

Keyword(s):

Disjoint Union ◽

A Priori ◽

Holomorphic Functions ◽

Stable Rank ◽

Open Unit ◽

Uniform Estimates ◽

Approximation Theorems ◽

Type Approximation ◽

Similar Approximation ◽

Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-021-09394-2 ◽

2021 ◽

Vol 24 (3) ◽

Author(s):

Alexander I. Bobenko ◽

Ulrike Bücking

Keyword(s):

Error Estimate ◽

Branch Point ◽

Riemann Surfaces ◽

Riemann Sphere ◽

Holomorphic Functions ◽

Convergence Result ◽

Edge Length ◽

Ramified Covering ◽

Compact Riemann Surfaces ◽

Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Univalence Criteria for Holomorphic Functions Involving Srivastava-Attiya Operator

Journal of Physics Conference Series ◽

10.1088/1742-6596/1879/2/022130 ◽

2021 ◽

Vol 1879 (2) ◽

pp. 022130

Author(s):

Huda F. Hussian ◽

Abdul Rahman S. Juma

Keyword(s):

Holomorphic Functions ◽

Univalence Criteria