A First Course in Complex Analysis with Applications by Dennis G Zill is a widely used textbook that provides a comprehensive introduction to the subject of complex analysis. This book is designed for students studying mathematics at the undergraduate level and aims to provide a solid foundation in the concepts and techniques of complex analysis.
The book covers a wide range of topics, from the basics of complex numbers and the complex plane to more advanced topics such as derivatives, complex integration, and conformal mapping. The author, Dennis G Zill, is known for his clear and accessible writing style, which makes the material easy to understand for students with varying levels of mathematical background.
Table of Contents
Complex Number and the Complex Plane.
Complex Number and Their Properties . . . . .
Complex Plane . . . . . .
Polar Forms of Complex Number . . . . .
Powers and Roots
Sets of Points in the Complex Plane
Application
Complex Functions as Mappings
Comlex Functions . . . .
Complex Functions as Mappings . . . . . .
Linear Mappings . . . . . .
Special Power Function . . . . . .
Reciprocal Function . . . . . .
Limits and Continuity . . . . .
Applications
Analytics Functions
Differentiability and Analyticity . . . . .
Cauchy-Riemann Equations . . . . . .
Harmonic Functions . . . . . . .
Applications . . . . . . . .
Elementary Functions
Exponential and logarithmic Functions . . . . . . . . .
Complex Powers . . . . .
Trigonometric and Hyperbolic Functional . . . . . . .
Inverse Trigonometric and Hyper bolic Functional . . . . . .
Applications . . . . . . .
Integration in the Complex Plane
Real Integrals . . . . .
Complex Integrals . . . . .
Cauchy-Goursat Theorem . . . . .
Independence of path . . . . .
Cauchy,s Integral Formulas and their Consequence . . . . . .
Applications . . . . . . .
Series and Residues
Sequence and series . . . . . .
Taylor Series . . . . . .
Laurent Series . . . .
Zeros and Poles . . . . . ..
Residues and Residue Theorem . . . . . . . .
Some Consequence of the Residue Theorem . . . . . . . . .
Applications
Conformal Mappings
Conformal Mappings . . . . . .
Linear Fractional Transformations. . . . . . .
Schwarz-ChristoffelTransformations . . .. . . . .
Poisson Intergral Formulas . . . . .
Applications . .. . . .. . . . .
One of the strengths of A First Course in Complex Analysis with Applications is its focus on applications, which helps students to see the practical applications of complex analysis and understand how the concepts covered in the book can be used in real-world situations. Additionally, the book includes numerous examples and exercises, which provide opportunities for students to practice what they have learned and apply the concepts covered in the book to new problems.
Whether you are a student looking to build a solid foundation in complex analysis or a professional seeking to refresh your knowledge of the subject, A First Course in Complex Analysis with Applications by Dennis G Zill is an excellent resource that is sure to meet your needs. With its comprehensive coverage of the subject and its focus on applications, this textbook is sure to provide a valuable resource for anyone interested in learning about complex analysis.
