scholarly journals The Cauchy-Length Formula and Holditch Theorem in the Generalized Complex Plane C_p

2018 ◽  
Vol 11 (2) ◽  
pp. 111-119
Author(s):  
Tülay Erişir ◽  
Mehmet Ali Güngör
Keyword(s):  
Complex Plane ◽  
Holditch Theorem ◽  
Generalized Complex

scholarly journals A New Construction of Holditch Theorem for Homothetic Motions in C p

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Tülay Erişir
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Number System ◽  
Geometric Representation ◽  
Area Formula ◽  
Holditch Theorem ◽  
Generalized Complex ◽  
New Construction ◽  
The One ◽  
The Relationship

In this study, the planar kinematics has been studied in a generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the planar kinematic formulas with one parameter for homothetic motions in the generalized complex plane have been mentioned briefly. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in a generalized complex plane have been obtained during the one-parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, has been expressed for homothetic motions in a generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals The Uniformisation of the Equation $$z^w=w^z$$

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Positive Solutions ◽  
Complex Plane ◽  
Complex Variable ◽  
Holomorphic Functions ◽  
Parametric Form ◽  
Lambert Function ◽  
The Real ◽  
Complex Equation ◽  
Real Equation ◽  
Expository Paper

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 .

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