Singularities of bicomplex holomorphic functions

2021 ◽  
Author(s):  
M. Elena Luna‐Elizarrarás ◽  
C. Octavio Perez‐Regalado ◽  
Michael Shapiro
Keyword(s):  
Holomorphic Functions ◽  
Bicomplex Holomorphic Functions

Bicomplex Holomorphic Functions

2015 ◽  
Author(s):  
M. Elena Luna-Elizarrarás ◽  
Michael Shapiro ◽  
Daniele C. Struppa ◽  
Adrian Vajiac
Keyword(s):  
Holomorphic Functions ◽  
Bicomplex Holomorphic Functions

The Cauchy-Kowalewski product for bicomplex holomorphic functions

2012 ◽  
Vol 285 (10) ◽  
pp. 1230-1242 ◽  
Author(s):  
H. De Bie ◽  
D. C. Struppa ◽  
A. Vajiac ◽  
M. B. Vajiac
Keyword(s):  
Holomorphic Functions ◽  
Bicomplex Holomorphic Functions

Some Properties of Bicomplex Holomorphic Functions

2015 ◽  
pp. 179-191
Author(s):  
M. Elena Luna-Elizarrarás ◽  
Michael Shapiro ◽  
Daniele C. Struppa ◽  
Adrian Vajiac
Keyword(s):  
Holomorphic Functions ◽  
Bicomplex Holomorphic Functions

On the Laurent series for bicomplex holomorphic functions

2017 ◽  
Vol 62 (9) ◽  
pp. 1266-1286 ◽  
Author(s):  
M. E. Luna-Elizarrarás ◽  
C. O. Pérez-Regalado ◽  
M. Shapiro
Keyword(s):  
Laurent Series ◽  
Holomorphic Functions ◽  
Bicomplex Holomorphic Functions

scholarly journals NORMAL FAMILIES OF BICOMPLEX HOLOMORPHIC FUNCTIONS

Fractals ◽  
2009 ◽  
Vol 17 (03) ◽  
pp. 257-268 ◽  
Author(s):  
K. S. CHARAK ◽  
D. ROCHON ◽  
N. SHARMA
Keyword(s):  
Complex Variable ◽  
Julia Set ◽  
Julia Sets ◽  
Holomorphic Functions ◽  
Normal Families ◽  
General Definition ◽  
Fatou Set ◽  
Definition Of ◽  
Bicomplex Holomorphic Functions

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice j = 0.

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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