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Complex Analysis and Applications

Written by Hemant Kumar Pathak

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940 pages, about 19 hours of reading

Mathematics

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About this book

This book offers an essential textbook on complex analysis. After introducing the theory of complex analysis, it places special emphasis on the importance of Poincare theorem and Hartog’s theorem in the function theory of several complex variables. Further, it lays the groundwork for future study in analysis, linear algebra, numerical analysis, geometry, number theory, physics (including hydrodynamics and thermodynamics), and electrical engineering. To benefit most from the book, students should have some prior knowledge of complex numbers. However, the essential prerequisites are quite minimal, and include basic calculus with some knowledge of partial derivatives, definite integrals, and topics in advanced calculus such as Leibniz’s rule for differentiating under the integral sign and to some extent analysis of infinite series. The book offers a valuable asset for undergraduate and graduate students of mathematics and engineering, as well as students with no background in topological properties.

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Reading guide

Themes, characters and key ideas in Complex Analysis and Applications, written by Chaptra AI.

  • about 400 hours
  • advanced
  • academic
  • rigorous
  • challenging

Hemant Kumar Pathak's "Complex Analysis and Applications" is a comprehensive and rigorous textbook designed for undergraduate and graduate students in mathematics and engineering. Spanning 940 pages, the book meticulously builds the foundational theory of complex analysis before delving into more advanced topics, with a particular emphasis on Poincare's Theorem and Hartog's Theorem within the function theory of several complex variables. It highlights the vast interdisciplinary relevance of complex analysis, connecting it to fields such as linear algebra, numerical analysis, geometry, number theory, various branches of physics, and electrical engineering. While requiring some prior knowledge of complex numbers and basic to advanced calculus, the book is structured to lay a solid groundwork for future specialized study.

“Not applicable in the literary sense. Key mathematical concepts and theorems emphasized include Poincare's Theorem and Hartog's Theorem, which are central to the function theory of several complex variables.”

Key themes

Fundamental Theory of Complex Analysis
The book comprehensively covers the core principles of complex analysis, including complex numbers, analytic functions, contour integration, residues, series representations, and conformal mappings. This forms the essential bedrock upon which all subsequent advanced topics are built.
Function Theory of Several Complex Variables
A significant focus of the book is on extending complex analysis to multiple complex variables, with particular attention to Poincare's Theorem and Hartog's Theorem. This theme explores the unique challenges and profound differences that arise when moving from one to several complex dimensions, highlighting key results that do not have direct analogues in real analysis.
Interdisciplinary Applicability of Complex Analysis
The book consistently emphasizes the wide-ranging applications of complex analysis across various scientific and engineering disciplines. This theme demonstrates the practical utility and foundational role of complex analytical techniques in solving real-world problems, from fluid dynamics to electrical circuit analysis.

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