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## Meet the editors

Dr. Francisco Bulnes, PhD, PostDocs, Doctor H. C., HonDSc, zbMATH, MathSci PhD in Mathematical Sciences, IM/UNAM. IINAMEI Director, Mathematics Research Centre in Mexico, 2015-present. Editor-in-Chief of Journals of Mathematics, in USA, and India, 2015-present. Member of various international committees of science. Reviewer of British journals of Mathematics and Physics in SCOPUS; Head of Research Department,

GI-TESCHA. Numerous papers (more than 100) in mathematics and physics research journals, and author of several books of mathematics and physics. Recognized in East Europe, Asia, Arab continents. He has many theories, theorems, and math objects with his name. He has received various honors and awards (Doctorates Honoris Causa) by universities and NGOs, likewise GOs. He received the Doctor Honoris Causa in Education Philosophy and Peace Ambassador by ODAEE in Frankfurt, Germany. He is also a Czech Republic Mathematics Society distinguished member (JCFM). He has two post-doctorates in Cuba and Russia in mathematics. Many international awards and badges (more than 50) as Publons badge, SCOPUS, ZbMath, Thomsom Reuters, ORCID, Peace Ambassador and others. In addition, he has undertaken advanced research in electronics, micro-electronics and spintronics. He is an author, reviewer, book editor, and collaborator with IntechOpen from 2010.

Dr. Olga A. Hachay graduated with a degree in Astrophysics from Ural State University in 1969. She obtained her PhD from the Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences (IZMI-RAN) in 1979 with her thesis "The inverse problem for electromagnetic research of one-dimensional medium." Since 1969, she has been a scientific member of the Institute of Geophysics

Ural Branch of Russian Academy of Sciences (UB RAS), Ekaterinburg, Russia. From 1995 to 2004, she served as chief of the group of seismic and electromagnetic research. Her research interests include developing new methods for searching the structure and the state of the Earth's upper crust, as well as elaborating a new theory of interpretation of electromagnetic and seismic fields. From 2002, she has been the main scientific researcher of the Institute of geophysics UB RAS. Since 2008, she has been a lead scientific researcher for UB RAS in the laboratory of borehole geophysics. Dr. Hachay is a member of various organizations and societies, including the American Mathematical Society, Mathematical Association of America, International Association of Geomechanics, and the European Geosciences Union, among others. Dr. Hachay is fluent in Russian, English and German language.

Contents

Introductory Section

of the Complex Analysis *by Francisco Bulnes*

**Preface XI**

**Section 1 1**

**Chapter 1 3**

**Section 2 9**

**Chapter 2 11**

**Chapter 3 27**

**Chapter 4 41**

**Section 3 63**

**Chapter 5 65**

**Chapter 6 79**

Introductory Chapter: Frontiers and Future Developments

Complex Functions and Some of Their Functional Transforms

Interference Pattern Representation on the Complex s-Plane

Solution Methods of Large Complex-Valued Nonlinear System

Advanced Complex Analysis and Geometry Studies

Integral Geometry and Cohomology in Field Theory on the

Variable Exponent Spaces of Analytic Functions *by Gerardo A. Chacón and Gerardo R. Chacón*

Space-Time as Complex Riemannian Manifold

*by José Trinidad Guillen Bonilla, Alex Guillen Bonilla, Mario Alberto García Ramírez, Gustavo Adolfo Vega Gómez,*

*Héctor Guillen Bonilla, María Susana Ruiz Palacio,*

*Martín Javier Martínez Silva and Verónica María Bettancourt Rodriguez*

Inverse Scattering Source Problems *by Mozhgan "Nora" Entekhabi*

of Equations *by Robson Pires*

*by Francisco Bulnes*

### Contents



Preface

The complex analysis is known as the theory of functions of complex variables inside of the mathematical analysis having extensions in arithmetic and algebraic geometry, number theory, and other sophisticated theories as the Nevanlinna theory to the meromorphic functions study, the Morse theory and complex projective geometry, where this last arises from the hyperbolic geometry and their extensions. The present book is centered on the advances in holomorphicity, analyticity of complex functions of one or several variables, their integration and evaluation. From the point of view of their geometry, we discuss the conformal mappings of surfaces and complex Riemannian surfaces, considering their singularities and their treatment through diverse tools that offer the complex analysis theory. Also we focus on the topology relationships between geometrical manifolds considering the integration invariants and their cohomology. In addition, diverse applications in inverse problems and ill posed problems of complex functions in engineering fields

**Dr. Francisco Bulnes**

TESCHA, Chalco, Mexico

**Olga A. Hachay**

en Matemáticas e Ingeniería),

Yekaterinburg, Russian Federation

IINAMEI A. C. (Investigación Internacional Avanzada

Research Department in Mathematics and Engineering,

Ural Branch of the Russian Academy of Sciences,

will be important.
